What we learn from studying the speed of waves

1. Knowing the propagation speed of mechanical waves in a medium provides valuable information on the physical characteristics of the medium itself.
   • In the case of gases, the propagation speed of the wave can give indications on the mass of the molecules making up the gas, and on the different tendency of a gas to heat up when it is kept at constant volume or at constant pressure. Moreover, since it grows as the temperature grows, the speed of mechanical waves also provides an indirect check of the microscopic interpretation of temperature as the average kinetic energy of the particles.
   • In the case of solids and liquids, the speed of waves is closely related to the elasticity and density of the medium, and elastic waves are routinely used as a means of investigating the subsoil, for example to locate oil fields.
2. Another interesting aspect of knowing the propagation speed of the wave is more closely related to the possibility of exploiting the wave itself
   • to transfer energy from one place to another: for example, one could set in motion a floating object at some distance from us by generating waves on the surface of the water which, once they reach the object, set it in motion;
   • to transfer information from one place to another (in this sense the propagation speed measures how fast information is transferred);
   • to locate the epicentre of an earthquake by exploiting the different propagation speeds in solids (rocks) of two types of waves present at the same time (P waves and S waves);
   • to locate obstacles by exploiting the phenomenon of reflection of sound waves and calculating (the speed being known) the time interval between the emission of the wave and the detection of the reflected wave. This is the working principle of sonar (acronym of sound navigation and ranging)
   • to determine the speed at which an object is moving away or approaching by exploiting the so-called Doppler effect. Doppler sonar and Doppler radar (used by the traffic police as a speed camera) are based on this principle.
3. The last interesting aspect, and for us the most important given its connections with the production of sound, lies in the fact that knowing the propagation speed of waves in strings (struck, plucked or bowed), membranes, or air columns excited by a musician's breath makes it possible to know the pitch of the note produced by the various musical instruments.

A general view

• The ability to propagate, that is, to move in spacetime, is a general feature of all waves, so much so that it is part of the definition of a wave, whatever its nature.
• Mechanical, or elastic, waves are a particular type of wave in which the propagating disturbance is a local deformation of the medium. These waves can exist and propagate precisely thanks to the property of material bodies called elasticity. A medium is said to be elastic if, following a deformation, it develops internal forces that tend to restore its original shape and size. To a first approximation all material media have this property, at least in the limit of small deformations.
• There are, however, different ways of deforming a body (compression, bending, tension, torsion, etc.) and, consequently, different types of mechanical waves. The simplest classification distinguishes transverse and longitudinal waves, but torsional waves, surface waves, etc. are also possible (see the examples on the page on transverse and longitudinal waves).
• Moreover, the elasticity of a body generally varies with the type of deformation imposed on it. For example, a body can be very resistant to compression but very fragile under tension, etc....
• It follows that the different types of mechanical waves, even in the same medium, will have different propagation properties.

We have thus come to ask which properties of the medium the propagation speed of mechanical waves depends on.

There is a very general answer to this question: the propagation speed of elastic waves depends on only two characteristics of the medium: its elasticity (with respect to the type of wave under consideration), and its density.

Let us try to understand why using a simple model.

A simple model

• Imagine having a chain of small spheres of mass m, equally spaced and connected by springs. Initially all the masses are subject to balanced forces (that is, with zero resultant), and are at rest.
• Imagine pushing the first sphere towards the chain by a length ${\displaystyle \mathrm{\Delta }x}$ while keeping all the other spheres fixed in place (which amounts to a compression). By the law of linear elasticity (or Hooke's law), the force that develops on the sphere (and, by the principle of action and reaction, also on the rest of the chain) is

${\displaystyle F=-k\mathrm{\Delta }x}$,

where k is a constant of proportionality, with dimensions N/m (that is, kg/s²), called precisely the elastic constant. The minus sign indicates that the force opposes the deformation, in the sense that it tends to restore the initial position of the element.

• By the second law of dynamics (Newton's law of motion), however, a force F applied to a mass m gives rise to an acceleration

${\displaystyle a=-k\frac{\mathrm{\Delta }x}{m}}$.

If, from now on, all the masses are free to move, they will undergo, one after the other, a force that depends on the displacement of the neighbouring masses.

• Note that, in this model, it is not possible for all the masses to start moving instantaneously together with the first. Indeed, even though the Newtonian force between two successive masses applies instantaneously (it depends only on ${\displaystyle \mathrm{\Delta }x}$), we see that the acceleration of the second mass is directly proportional to k and inversely proportional to m. This means that, for the same initial compression ${\displaystyle \mathrm{\Delta }x}$, the second mass will move the faster the stiffer the springs (k is large), or the "lighter" the masses (m is small).
• The faster the masses manage to move away from their equilibrium position, the shorter the time needed for the next mass to undergo in its turn a displacement ${\displaystyle \mathrm{\Delta }x}$. In a word, the faster the compression will propagate along the chain.

Once we have understood this, we have grasped the point. We are not so much interested in the complete solution of the motion of each individual sphere (an oscillating solution), as in the fact that only two quantities characteristic of the medium can contribute to defining the speed of the compression wave. They are precisely k and m.

• Had the wave been of a nature other than compression, the reasoning could still be applied, changing the quantity that defines the deformation (${\displaystyle \mathrm{\Delta }x}$) and using an appropriate elastic constant k. So the reasoning is generally valid.
• If the medium, instead of being a discrete set of masses, is described as a continuum, the single mass m and the single stiffness k become infinitesimal, while the density ρ (measured in kg/m³) and the elastic constant per unit length κ, measured in Pa (that is, it has the dimensions of a pressure, 1Pa = 1 N/m²), remain finite.
• The only combination of these two quantities that has the dimensions of a speed is

${\displaystyle v\propto \sqrt{\frac{\kappa }{\rho }}}$.

Dimensional analysis, of course, can define a quantity only up to dimensionless constants. This explains why we have not used the "equals" sign, but the "proportional" one.

Speed of the different types of waves

In this section we take the general reasoning of the previous section and apply it to the different types of elasticity in the different media.

Waves on a stretched string

The simplest case, and the most important one for music, is that of a stretched string, like those of string instruments. Here the restoring force does not come from the elasticity of the material but from the tension of the string: if a stretch of string is displaced sideways, the tension pulls it back towards the equilibrium position. In place of the density we have the mass per unit length of the string. The waves are transverse and their speed is

Tightening a string makes the wave faster and the note higher; at equal tension, heavier strings (such as bass strings) are slower and give lower notes (see normal modes of a string and wave equation on a string).

Waves in gases

In gases and vapours the molecules are essentially free to move, without being bound to each other by strong forces, unlike what happens in solids. Indeed, we can change the shape of a gas without it offering any resistance.

However, a gas shows elastic behaviour when it is compressed. Indeed, by confining the molecules in smaller and smaller spaces, the pressure they exert on the walls of the container holding the gas increases and then, when the spaces become even smaller, they begin to interact with each other.

Consequently, the only waves that can propagate in a gas are compression waves. When these waves have frequencies in the range of about 20-20000 Hz, they are called sound waves, and so for the discussion we refer to the page on the speed of sound.

Here we only report the result:

While it is fairly intuitive that pressure plays the role of elasticity, it is not obvious why the factor γ should appear in the formula. It is equal to the ratio between the specific heats at constant pressure and at constant volume, and is always greater than 1. See the discussion on this subject on the page speed of sound.

Waves in liquids

In liquids too the molecular bonds are rather weak, and liquids therefore have no shape of their own. Ideally, the molecules of a liquid can slide freely in parallel planes, that is, the sliding of the planes is not counterbalanced by a restoring force. Liquids therefore have no elasticity for shear deformations either and, again, inside an infinite fluid only longitudinal (or compression) waves can propagate. In real cases, of course, there are friction forces (e.g. viscous friction) acting between one sliding plane and another. These forces, however, are typically much weaker than the cohesion of the liquid within each plane, and here we neglect their effect.

Liquids, like gases, have an elasticity for compression, but they are much stiffer than gases. Water, for example, is about ten thousand times less compressible than air, which results in a bulk modulus about 10000 times larger.

Apart from this substantial quantitative difference, the speed of waves in liquids is given by a formula analogous to the one given for gases.

The thermodynamic factor γ does not appear in this case because, precisely owing to their enormous stiffness, in liquids (as in solids) there is no substantial difference between the specific heat at constant volume and at constant pressure.

New types of waves at the surface

The compression waves examined in the previous section are the equivalent of sound waves in liquids. They are particularly important in underwater applications, so much so that ocean acoustics is a specialized branch. However, the most familiar type of waves in liquids are the waves we see, for example, on the surface of the sea. These waves must not be confused with compression waves, because they are a completely different phenomenon.

Surface waves are a particular type of wave that propagates at the separation surface between two different media. For example, at the surface of the sea, between water and air.

It is easy to see that these waves do not behave like ordinary compression waves. Surface waves, in fact,

• develop in a condition of reduced symmetry: the elastic forces and the densities of the two media in contact can be very different. The molecules at the surface of the water feel upward forces (exerted by the air) completely different from the downward ones (exerted by other water)
• propagate only along the separation surface, and do not penetrate appreciably into either medium.
• their speed depends not only on the characteristics of the two media, but also on characteristics of the waves. Typically on the wavelength.

Referring precisely to the case of water waves, we can identify two important approximations:

• Yet other types of waves are produced if, besides the effects of gravity and of the fluid pressure, the terms due to surface tension and capillarity are taken into account. In the case of water, these terms become important only when the dimensions of the volume of water under consideration are very small (or the gravitational field vanishes, as in experiments carried out in orbit). These effects are not dealt with here.

Waves in solids

In solids the cohesive forces between molecules are much stronger than in fluids. Indeed, solids have their own shape and volume. We can imagine a crystalline solid as a lattice of atoms bound together by forces of electrical nature. The forces act in all directions, so an arbitrary deformation of the body can give rise to a complex, distributed stress.

For example, a beam subject to its own weight and fixed at its ends tends to bend, curving in the middle. The resulting deformation is a compression on the upper face, but a tension on the lower face.

The theory of elasticity in solid bodies nevertheless allows us to generalize Hooke's law by saying that

in the elastic regime, stress is directly proportional to strain.

Moreover, we can classify deformations into only three main classes: compression, tension and shear.

• Compression is the same as for fluids: a decrease in volume caused by a force perpendicular to the surface on which it acts.
• Tension is produced when two forces act in opposite directions perpendicular to the surface. It causes the solid to lengthen.
• Shear is the deformation produced by a couple of forces applied tangentially to the surface. Shear stress occurs, for example, in torsion.

In any case, our general reasoning of the second section on the speed of mechanical waves keeps all its force. The interesting aspect of the question is that the elastic constants of a solid for these three types of deformation are generally completely different. So, in the case of solids, there is no such thing as the speed of waves, but rather the speeds of waves. In particular, three elastic constants are defined, one for each type of stress. All the constants are defined in the same way, as the ratio between stress and strain.

A table with purely indicative order-of-magnitude values. Note that 1 GPa = 10⁹ Pa. Note that some fibrous materials, such as wood, have very different moduli in the directions parallel and perpendicular to the fibres.

In the same medium, that is, at fixed density, these constants determine the speed of the three types of waves.

Transverse (or shear) waves

Transverse waves always correspond to shear stresses, because they tend to make the molecules of the solid slide along planes perpendicular to the direction of transmission of the wave.

Longitudinal (or compression) waves

For compression waves in solids a distinction must be made. Longitudinal waves in a solid have different propagation speeds depending on whether one considers a bar, that is, a portion of the solid whose transverse dimensions are much smaller than the wavelength, or a block of the solid, in which all the dimensions are much larger than the wavelength. In the case of a bar we could think of striking one end with a hammer and measuring the speed at which the resulting compression propagates. The case of the block always occurs deep inside the Earth for seismic waves, or for elastic waves in crystals.

Indeed, we have

The reason for this distinction lies in the fact that, in a bar under tension, besides an elongation there is a reduction of the cross-section (Poisson effect), because the cohesive forces tend to bring the atoms closer together in the direction perpendicular to the applied tension. This effect is easy to observe by stretching a piece of rubber with your hands. The same phenomenon occurs in the block too, but the magnitude of the lateral forces is completely different, because there is no boundary surface.

Surface waves

At the solid-air surface too, waves in the solid take on a new form and become surface waves. These waves are particularly important because they are among the ways the ground oscillates during earthquakes. Surface waves have a propagation speed different from that of both transverse and longitudinal waves. Indeed, this property can conveniently be exploited to determine the position of the epicentre of an earthquake.

Relations between the speeds, and seismic waves

In solids there is an approximate relation between the speeds of transverse and longitudinal waves. Indeed, of all the elastic moduli only two are independent, while simple algebraic relations exist among all the others. In particular

${\displaystyle Y=\frac{9KG}{3K+G}}$

In practice, for most rocks the bulk modulus K is about 5/3 of the shear modulus G, so ${\displaystyle K+\frac{4}{3}G\approx 3G}$: comparing the formulas for the block and for transverse waves, it follows that the speed of longitudinal waves is roughly ${\displaystyle \sqrt{3}\approx 1.7}$ times that of transverse waves.

In general, therefore, following a mechanical disturbance in a solid, we will always obtain at least two wave trains that reach a distant observer at different times. This observation is so important for geophysics that it requires a specific terminology:

• longitudinal waves are also called P waves (or primary waves)
• transverse waves are also called S waves (or secondary waves).

The delay between the arrival of a P wave and an S wave can usefully be used to locate the epicentre of an earthquake. If a seismograph records the difference ${\displaystyle \mathrm{\Delta }t}$ between the arrival times of the S waves and the P waves, calling D the distance of the seismograph from the epicentre of the earthquake and denoting by ${\displaystyle v_p}$ and ${\displaystyle v_s}$ the speeds of the primary and secondary waves, it is easy to see that

${\displaystyle \mathrm{\Delta }t=\frac{D}{v_s}-\frac{D}{v_p}}$,

from which

${\displaystyle D=\mathrm{\Delta }t\cdot \frac{v_p\cdot v_s}{v_p-v_s}}$.

Obviously, with only one seismograph it is not possible to determine the epicentre of the earthquake: knowing D only tells us that the epicentre lies on a sphere of radius D centred at the point where the seismograph is. With a network of seismographs (three are enough), however, it is possible to "intersect" the various spheres centred on the different seismographs to obtain the only point that can be the epicentre of the earthquake.

But the Earth's core is liquid!

The absence of S waves in fluids has a very important application in geophysics: if a seismograph detects only P waves, it means that the S waves have met a "liquid" zone unable to transmit them. By comparing the "shadow zones" of the various seismographs scattered around the Earth, it has been possible to work out the size of the Earth's outer core, which is liquid: its radius is roughly half that of the Earth. Inside it there is a smaller inner core, which is solid.

"Unorthodox" examples

The stadium wave

Technically this is not a purely elastic wave, because it cannot be described in terms of just two mechanical parameters such as k and m. In fact the characteristics of the wave also depend on quantities such as the excitability of the fans, the importance of the championship, etc. However, it is not difficult to formulate a simple physical model to try to calculate its propagation speed. Let us first make a naive attempt.

Given that the wave is defined by the propagation of the height of the spectators with respect to the level of the seated spectators, we could, for example, think that the contact mechanism works like this: each person tries to stay at the same height as their right-hand neighbour. So each person stands up as quickly as possible on seeing the neighbour stand up, and sits down again as quickly as possible on seeing the neighbour sit down. In this case the phase delay between two neighbouring spectators is determined only by the reaction time, while the speed with which each of them goes from the "seated" to the "standing" state depends on the mass and on the elasticity (that is, the muscle strength) of the spectator. Using "standard spectators", all with the same mass and muscle strength, and with equally spaced seats, we have an approximation corresponding to the homogeneous medium (also used in the chain model of the previous section). Estimating a reaction time of 0.1 s, with a distance of 60 cm between one seat and the next, we would have a speed of 6 m/s.

The observed speed is around 12 m/s, which clearly means that other parameters must be taken into account, and the simple mechanical model does not apply to this type of wave. One of the reasons that explain the higher propagation speed of the real wave is the fact that the interaction between people at the stadium does not occur only on the scale of nearest neighbours, but on a larger scale. Indeed, the wave does not generally have an amplitude spread over the width of one seat (as we have implicitly assumed), but over some fifteen seats.

The example may seem an end in itself, but modelling propagation in excitable media is very important for studying the propagation of forest fires and of electrical waves in heart muscle. It seems that a crowd of fans can likewise be considered an "excitable medium" (see the article published in the journal Nature^[1]). Moreover, on the website of Eötvös University in Budapest you will find an interactive simulation based on the mathematical model proposed for the stadium wave.

The queue at the traffic light

The traffic wave has a particularly irritating effect when, being in a queue some distance from a green traffic light, you remain stuck until the light turns red, and only then can you start moving, only to have to stop shortly afterwards. The phase delay between the colour of the traffic light and the arrival of the compression wave of the distance between cars directly measures the propagation speed of this wave, given the distance from the traffic light.

Another observable case is the so-called traffic "slowdowns" on a motorway. The flow of cars is similar to the flow of a compressible medium and, indeed, the average distance between cars can vary. A slowdown corresponds not only to a decrease in the speed of the vehicles, but also to a sudden compression of the "fluid": in a slowdown the average distance between vehicles decreases to the minimum allowed, and propagates backwards roughly as a sound wave would in an elastic medium.

In this case too, the medium (the column of cars) cannot simply be considered subject to Hooke's law. However, a certain similarity is evident: the "force" that keeps vehicles apart is not a physical force, but is implicitly produced by driving behaviour: one tends to slow down when getting "too" close to the vehicle in front, and to accelerate when the distance is large.

In this case too, simply measuring reaction times may describe the phenomenon well (as in the case of the queue at the traffic light), or not (as in the case of the motorway slowdown).

Further reading

For a more rigorous and thorough derivation of elastic waves, see the many pages on the wave equation and, in particular, wave equation in a chain of oscillators for the analysis of the chain model.

Besides the characteristics of the medium, some characteristics of the wave can also affect the propagation speed. In particular it can depend on amplitude, frequency and polarization. These less elementary aspects will be examined in part on the pages devoted to dispersion and to non-linear effects. Dispersion in particular has important consequences in determining the sound quality of many musical instruments.

In fact, it is important to realize that the formulas given above for the propagation speeds (derived in the sections wave equation in a string and wave equation in a bar) are approximate formulas, valid only

1. for small displacements from the equilibrium position
2. in the absence of dissipative phenomena that convert the mechanical energy of deformation into other forms (electrical, magnetic, thermal, nuclear, etc.)

The first hypothesis expresses Hooke's law (ut tensio, sic vis), and so allows waves to be treated as a linear phenomenon, excluding shock waves and anharmonic forces. The second hypothesis simply goes beyond the field of pure mechanics, and requires knowledge of other branches of physics.

To conclude, examples of mechanical waves, besides sound waves, to which a dedicated section is devoted, are

• the vibrations of mechanical systems (see resonance in bridges, resonance in buildings)
• the shock waves produced in detonations, in collisions, or in supersonic fluids (see the loudest sounds, and beyond the sound barrier). The different waves may or may not obey the superposition principle, depending on the case.

Check your understanding

Further reading and links

• If you want to look more deeply, from the mathematical point of view, at the derivation of the speed of waves in the various cases, visit the section Wave equation;
• If you want to know in particular the data on the speeds of sound waves in the various media, go to the page speed of sound in media;
• To visualize other types of waves (intermediate between transverse and longitudinal ones), visit the section Transverse and longitudinal waves;
• For more on sound waves, visit the section speed of sound, where curious experiments "in helium" are described;
• On the page on the Kundt tube you will find an experiment to measure the Young's modulus Y of a bar;
• To look further into the connections between the propagation speed of waves in musical instruments and the frequency of the note produced, visit the sections on musical instruments and natural frequencies.