In a pipe or sound tube the vibrating medium is air. The wave equation governing the acoustics of pipes is therefore the equation of sound waves.

This equation was derived on the page wave equation for sound, precisely in its one-dimensional form, which is particularly suited to pipes. Indeed, even though radial modes are allowed in pipes, they always have much higher frequencies than the longitudinal modes, since the bore of musical instruments is always much longer than it is wide. The frequencies of the radial modes therefore always lie well above the frequencies the instrument is tuned to produce.

On this page we want to show the differences in the form of the wave equation between a cylindrical and a conical pipe. There are musical instruments based on both these shapes, and their acoustic characteristics ultimately depend on these equations.

Cylindrical pipe

The wave equation

If we neglect the radial modes, the wave equation for the sound pressure p

${\displaystyle {\mathrm{\nabla }}^{2}p=\frac{1}{c^2}\frac{{\partial }^{2}p}{\partial t^2}}$

(1)

becomes the ordinary wave equation in its one-dimensional form along the axis x of the pipe

${\displaystyle \frac{{\partial }^{2}p}{\partial x^2}=\frac{1}{c^2}\frac{{\partial }^{2}p}{\partial t^2}}$.

The travelling solutions of this equation have the form

${\displaystyle p(x,t)=f(x-ct)+g(x+ct)}$.

Propagation therefore takes place through plane wavefronts, confined within the walls of the tube.

Standing waves

The normal solutions (that is, those in which all points vibrate at the same angular frequency ${\displaystyle \omega }$) are linear combinations of sines and cosines

${\displaystyle p_{\omega }(x,t)=[A\cos (kx)+B\sin (kx)]\sin (\omega t+\varphi )}$

k is the so-called wavenumber (in one dimension it coincides with the wave vector). It is related to the wavelength by

${\displaystyle k=\frac{2\pi }{\lambda }}$,

and, consequently, to the angular frequency by

${\displaystyle \omega =kc}$,

where c is the speed of sound in air.

The constants A, B and phi are determined from the initial and boundary conditions. To know instead whether a given frequency is a possible solution, we must check that the corresponding wave vector k is compatible with the boundary conditions.

For example, if the pipe is open at both ends, we must impose a pressure node at the ends. If the pipe starts at ${\displaystyle x=0}$ and ends at ${\displaystyle x=L}$, this is equivalent to imposing

${\displaystyle p(0,t)=p(L,t)=0}$,

from which ${\displaystyle A=0}$, and

${\displaystyle \sin (kL)=0}$,

that is

${\displaystyle k=\frac{n\pi }{L}}$,

where n is any integer. This determines the normal frequencies. They are integer multiples of a fundamental frequency

${\displaystyle \omega =n{\omega }_0}$,

with

${\displaystyle {\omega }_0=\frac{\pi c}{L}}$,

that is, the fundamental frequency is ${\displaystyle f_0=\frac{c}{2L}}$.

To see animations of the first normal modes in a cylindrical pipe, see the page normal modes of a cylindrical pipe

Pipe closed at one end

If instead the pipe is closed at ${\displaystyle x=0}$ and open at ${\displaystyle x=L}$, as to a first approximation in the clarinet, the air is at rest at the closed end. Since the acceleration of the air is proportional to ${\displaystyle -\partial p/\partial x}$, the pressure must have an antinode there:

${\displaystyle \frac{\partial p}{\partial x}(0,t)=0}$,

so that ${\displaystyle B=0}$. At the open end there is still a pressure node, ${\displaystyle p(L,t)=0}$, so

${\displaystyle \cos (kL)=0}$,

that is

${\displaystyle k=\frac{(2n-1)\pi }{2L}}$,

where n is a positive integer. The normal frequencies are therefore only the odd multiples of a fundamental

${\displaystyle {\omega }_0=\frac{\pi c}{2L}}$,

that is ${\displaystyle f_0=\frac{c}{4L}}$: an octave below the fundamental of the open pipe of the same length.

Conical pipe

The wave equation

The wave equation for pressure is still the ordinary wave equation, but this time a representation along the x axis alone is no longer enough to describe the physical system, since the pipe has a diameter that varies along the axis.

We can note, however, that portions of spherical waves propagate in the conical section, so the most suitable coordinates for our wave equation are spherical coordinates. If we again want to neglect the modes of propagation not parallel to the axis of the pipe, we can write the equation for spherical waves and consider only its radial part (that is, longitudinal with respect to the axis of the pipe). We put the origin of the coordinate system at the vertex of the cone.

Noting that

${\displaystyle {\mathrm{\nabla }}^{2}p=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial p}{\partial r}\right)=\frac{1}{r}\frac{{\partial }^2(rp)}{\partial r^2}}$,

and substituting into (1), after a few steps we find that, in spherical coordinates, the radial part of the wave equation is

${\displaystyle \frac{{\partial }^2(rp)}{\partial r^2}=\frac{1}{c^2}\frac{{\partial }^2(rp)}{\partial t^2}}$

(2)

.

In other words we still have the same type of equation, but the quantity it describes is no longer directly the sound pressure p, but the product of the sound pressure and the radius r, which is nothing but the distance from the vertex of the cone.

This means that the travelling solutions in the cone will have the form

${\displaystyle p(r,t)=\frac{f(r-ct)}{r}+\frac{g(r+ct)}{r}}$.

Standing waves

Equations (1) and (2) are therefore formally identical, but refer to different quantities and variables. We deduce that the normal solutions of (2) are identical to those of (1) once the appropriate substitutions have been made

${\displaystyle p_{\omega }(r,t)=[A\frac{\cos (kr)}{r}+B\frac{\sin (kr)}{r}]\sin (\omega t+\varphi )}$

Now it is not obvious how to impose the boundary condition at the vertex of the cone, since there ${\displaystyle r=0}$, and so, strictly speaking, the solution just written is not defined.

To find the correct solution we can proceed as follows: instead of a conical pipe, let us consider a truncated cone obtained by removing a small cone of slant height ${\displaystyle r_0}$ from the vertex of the cone.

In this case the solutions are of the type

${\displaystyle p_{\omega }(r,t)=[A\frac{\cos (k(r-r_0))}{r}+B\frac{\sin (k(r-r_0))}{r}]\sin (\omega t+\varphi )}$,

and are well defined at ${\displaystyle r=r_0}$, where we can impose the known condition of an open pipe, that is, a pressure node

${\displaystyle p(r_0,t)=p(L,t)=0}$.

We then again have ${\displaystyle A=0}$ and

${\displaystyle \frac{\sin (k(L-r_0))}{L}=0}$,

which again implies

${\displaystyle k=\frac{n\pi }{L-r_0}}$.

In complete analogy with the case of the cylindrical pipe, the normal frequencies of the truncated conical pipe are integer multiples of a fundamental

${\displaystyle \omega =n{\omega }_0}$,

with

${\displaystyle {\omega }_0=\frac{\pi c}{L-r_0}}$.

The fundamental is identical to that of a cylindrical pipe open at both ends whose length equals the height of the truncated cone.

To see animations of the first normal modes in a conical pipe, see the page normal modes of a conical pipe

The paradox of the conical pipe

Although we have seen that the solutions in spherical coordinates are not defined at the origin, this singularity is of course not physical; it comes only from the particular choice of coordinates, and shows up in the factor r appearing in the denominator of the solutions. This does not mean that the real pressure and velocity are not defined at the origin of a closed pipe. The value of the velocity is of course 0, because the air is always at rest at the point of contact with a fixed wall. As for the pressure, we can obtain its value at the origin, for example, by using the solutions for the truncated cone and taking the limit ${\displaystyle r_0\to 0}$. Indeed, since ${\displaystyle \underset{x\to 0}{\lim }\frac{\sin (x)}{x}=1}$ the limit of the solutions exists and is finite. The pressure at the vertex of the cone therefore has a finite value, as it is physically sensible to expect. Here, however, the paradox arises: in taking the limit, the boundary condition at the vertex changes: we have gone from an open end, with a pressure node, to a closed vertex, with a pressure antinode. In a cylindrical pipe, going from an open end to a closed one changes the natural frequencies radically. Why does this not happen in the conical pipe? The paradox arises because the functions describing the pressure in a complete conical pipe resemble those for a half-open cylindrical pipe. However, the resemblance is only superficial. Indeed, carefully analysing the pressure and velocity diagrams for the normal modes (the figures alongside show, for example, the third mode for all types of pipe), we notice that The positions of the pressure nodes of the truncated conical pipe, of the cylindrical pipe open at both ends, and of the complete cone are exactly the same. This means that the oscillation of the air in these three pipes has exactly the same wavelength, even though the specific shape of the modes is different. The positions of the pressure nodes of the closed-open cylindrical pipe are different from those of the other pipes, and in particular the wavelength is longer.

element red = sound pressure; green = air velocity wavelength Truncated cone ${\displaystyle \frac{2L}{3}}$ Open-open cylinder ${\displaystyle \frac{2L}{3}}$ Complete cone ${\displaystyle \frac{2L}{3}}$ Closed-open cylinder ${\displaystyle \frac{4L}{5}}$

In the discussion above we used the third mode only because it is easier to visualise, but if we now use the fundamental (that is, the first) mode for all the pipes we obtain wavelengths of ${\displaystyle 2L}$ and ${\displaystyle 4L}$ respectively, that is, frequencies in the ratio 1:2.

We conclude that the resemblance at the origin between the modes of the closed-open cylindrical pipe and those of the complete conical pipe is due only to the presence of the factor 1/r in the modes of the conical pipe. This factor appears in the conical case because the waves propagating in the pipe are spherical, not plane. If, however, we are interested in the emitted frequency, then it is the position of the nodes that determines the frequency of the mode, and the position of the nodes is the same in the conical, truncated conical and open-open cylindrical pipes, while it is completely different in the open-closed cylindrical pipe. It follows that this last type of pipe is tuned an octave below the others, which are instead all tuned to the same note.

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