Interference of harmonic waves

Imagine two point sources ${\displaystyle S_1}$ and ${\displaystyle S_2}$ that emit periodic waves with the same amplitude, frequency and phase.

Imagine also that the oscillatory motion of the two point sources ${\displaystyle S_1}$ and ${\displaystyle S_2}$ is "harmonic", that is, expressed by a mathematical function of the type

${\displaystyle y_{S_1}(t)=y_{S_2}(t)=A\sin (\omega t+\varphi )}$

(for a quick introduction to trigonometric functions see the page the sine and cosine functions)

The total wave function at point P is therefore given by

${\displaystyle y(\mathbf{𝐏},t)=y_{S_1}(t-t_1)+y_{S_2}(t-t_2)}$

${\displaystyle =A\sin [\omega (t-t_1)+\varphi ]+A\sin [\omega (t-t_2)+\varphi ]}$

${\displaystyle =A\sin {\theta }_1(t)+A\sin {\theta }_2(t)}$

where we have set

${\displaystyle {\theta }_1(t)=\omega (t-t_1)+\varphi }$

${\displaystyle {\theta }_2(t)=\omega (t-t_2)+\varphi }$

Constructive interference of harmonic waves in phase

The condition for constructive interference is obtained simply by requiring that the difference (in absolute value) of the arguments ${\displaystyle {\theta }_1(t)}$ and ${\displaystyle {\theta }_2(t)}$ be a multiple of ${\displaystyle 2\pi }$. We then obtain

${\displaystyle |{\theta }_1(t)-{\theta }_2(t)|=\omega |t_1-t_2|=2k\pi }$ with ${\displaystyle k=0,1,2,\dots }$ (1)

From (1) we deduce that the condition for constructive interference is independent of the time t and depends only on the position of point P (uniquely determined by the delay times ${\displaystyle t_1}$ and ${\displaystyle t_2}$). Moreover, since ${\displaystyle \omega =\frac{2\pi }{T}}$, (1) becomes

${\displaystyle \omega |t_1-t_2|=\frac{2\pi }{T}|t_1-t_2|=2k\pi }$

from which

${\displaystyle |t_1-t_2|=k\cdot T}$ with

${\displaystyle k=0,1,2,\dots }$

(1*)

which follows the condition in terms of time already derived in the general conditions for interference.

Since ${\displaystyle t_1=d_1/v}$ and ${\displaystyle t_2=d_2/v}$, where ${\displaystyle d_1}$ and ${\displaystyle d_2}$ are the distances of P from the two sources and v is the speed of the wave, and since ${\displaystyle vT=\lambda }$, the same condition can be written in terms of path difference:

${\displaystyle |d_1-d_2|=k\lambda }$

that is, interference is constructive at the points whose distances from the two sources differ by a whole number of wavelengths.

Destructive interference of harmonic waves in phase

The condition for destructive interference is obtained simply by requiring that the difference (in absolute value) of the arguments ${\displaystyle {\theta }_1(t)}$ and ${\displaystyle {\theta }_2(t)}$ be an odd multiple of ${\displaystyle \pi }$. We then obtain

${\displaystyle |{\theta }_1(t)-{\theta }_2(t)|=\omega |t_1-t_2|=(2k+1)\pi }$ with

${\displaystyle k=0,1,2,\dots }$

(2)

From (2) we deduce that the condition for destructive interference is independent of the time t and depends only on the position of point P (uniquely determined by the delay times ${\displaystyle t_1}$ and ${\displaystyle t_2}$). Moreover, since ${\displaystyle \omega =\frac{2\pi }{T}}$, (2) becomes

${\displaystyle \omega |t_1-t_2|=\frac{2\pi }{T}|t_1-t_2|=(2k+1)\pi }$

from which

${\displaystyle |t_1-t_2|=(2k+1)\cdot \frac{T}{2}}$ with ${\displaystyle k=0,1,2,\dots }$, (2*)

which follows the condition in terms of time already derived in the general conditions for interference. In terms of path difference:

${\displaystyle |d_1-d_2|=(2k+1)\frac{\lambda }{2}}$

that is, interference is destructive at the points whose distances from the two sources differ by an odd number of half wavelengths.

Interference of out-of-phase harmonic waves

Imagine that the two point sources ${\displaystyle S_1}$ and ${\displaystyle S_2}$ emit harmonic waves with the same amplitude and frequency, but not synchronised. Assume for example that source ${\displaystyle S_2}$ reproduces the oscillation of source ${\displaystyle S_1}$ with a certain time shift, which we will denote by ${\displaystyle t_s}$. We will take

${\displaystyle t_s>0}$ if source ${\displaystyle S_2}$ oscillates later than ${\displaystyle S_1}$

${\displaystyle t_s<0}$ if source ${\displaystyle S_2}$ oscillates earlier than ${\displaystyle S_1}$

If the motion of source ${\displaystyle S_1}$ is described by the function

${\displaystyle y_{S_1}(t)=A\sin (\omega t+\varphi )}$,

then the motion of source ${\displaystyle S_2}$ is described by

${\displaystyle y_{S_2}(t)=y_{S_1}(t-t_s)=A\sin (\omega (t-t_s)+\varphi )}$.

If, moreover, the time shift ${\displaystyle t_s}$ is a known fraction of the period T, that is

${\displaystyle t_s=\pm \alpha T}$ with ${\displaystyle \alpha \in ]0;1[}$

recalling, as usual, that ${\displaystyle \omega =\frac{2\pi }{T}}$, we obtain

${\displaystyle y_{S_2}(t)=A\sin [\omega (t\mp \alpha T)+\varphi ]=A\sin (\omega t+\varphi \mp 2\pi \alpha )}$.

In practice the time shift introduces a phase angle ${\displaystyle 2\pi \alpha }$, which allows the conditions for constructive (1) and destructive (2) interference to be reformulated at once.

Setting

${\displaystyle {\theta }_1(t)=\omega (t-t_1)+\varphi }$

${\displaystyle {\theta }_2(t)=\omega (t-t_2)+\varphi \mp 2\pi \alpha }$

condition (1) becomes

${\displaystyle |{\theta }_1(t)-{\theta }_2(t)|=|\omega (t_1-t_2)\pm 2\pi \alpha |=2k\pi }$ with

${\displaystyle k=0,1,2,\dots }$

(1**)

and condition (2) becomes

${\displaystyle |{\theta }_1(t)-{\theta }_2(t)|=|\omega (t_1-t_2)\pm 2\pi \alpha |=(2k+1)\pi }$ with

${\displaystyle k=0,1,2,\dots }$

(2**)

Interference between sources of different frequencies

All the previous interference conditions were derived under the assumption that the two interfering sources had the same frequency. The resulting conditions are independent of time and depend only on the position of point P in space.

If the sources do not oscillate at the same frequency, the interference conditions at a given point in space evolve in time, which makes the study of the phenomenon extremely complex. Suppose we have two "harmonic" sources ${\displaystyle S_1}$ and ${\displaystyle S_2}$ (with the same amplitude and phase) of different frequency. The oscillatory motion of the two sources can, as usual, be expressed by

${\displaystyle y_{S_1}(t)=A\sin ({\omega }_{1}t+\varphi )}$,

${\displaystyle y_{S_2}(t)=A\sin ({\omega }_{2}t+\varphi )}$,

with ${\displaystyle {\omega }_1\ne {\omega }_2}$; repeating the derivation carried out, for example, in the case of constructive interference, condition (1) now becomes

${\displaystyle |{\theta }_1(t)-{\theta }_2(t)|=|({\omega }_1-{\omega }_2)t-{\omega }_1{t}_1+{\omega }_2{t}_2|=2k\pi }$ with

${\displaystyle k=0,1,2,\dots }$

(1***)

which, as can be seen, depends on the time t.

Beats

If the two sources of different frequencies are located at the same point in space, it is clear that, for a fixed point P, the delay times ${\displaystyle t_1}$ and ${\displaystyle t_2}$ coincide. If we set ${\displaystyle t_1=t_2=\frac{x}{v}}$, the interference condition (1***) remains time-dependent, but takes the simpler form:

${\displaystyle |{\theta }_1(t)-{\theta }_2(t)|=|({\omega }_1-{\omega }_2)(t-\frac{x}{v})|=2k\pi }$ with

${\displaystyle k=0,1,2,\dots }$

(1°)

which can be written as:

${\displaystyle |t-\frac{x}{v}|=\frac{2k\pi }{|({\omega }_1-{\omega }_2)|}=k\frac{T_1\cdot T_2}{|(T_1-T_2)|}}$ with

${\displaystyle k=0,1,2,\dots }$

(1°°)

where, as usual, ${\displaystyle T_1}$ and ${\displaystyle T_2}$ denote the periods of the two sources. This condition gives all the instants of time t at which, at point P, at distance x from the two sources, there is constructive interference. If we consider two consecutive instants (that is, taken for values k and k+1) and evaluate their difference ${\displaystyle \mathrm{\Delta }t}$

${\displaystyle \mathrm{\Delta }t=\frac{T_1\cdot T_2}{|(T_1-T_2)|}}$

we realise that, for a fixed point in space, the interference condition recurs every ${\displaystyle \frac{T_1\cdot T_2}{|(T_1-T_2)|}}$ seconds, that is, it behaves periodically with frequency

${\displaystyle f=\frac{1}{\mathrm{\Delta }t}=\frac{|(T_1-T_2)|}{T_1\cdot T_2}=|f_1-f_2|}$

where, in the last step, the general relation between period and frequency was used, setting ${\displaystyle f_1=\frac{1}{T_1}}$ and ${\displaystyle f_2=\frac{1}{T_2}}$.

The frequency ${\displaystyle f=|f_1-f_2|}$ is called the beat frequency. By taking ${\displaystyle f_1}$ and ${\displaystyle f_2}$ almost equal, the beat frequency can be made much lower than the individual frequencies of the two interfering sources.

One last remark: the whole argument started from the condition for constructive interference. It remains valid, however, if one starts from condition (2) for destructive interference.

Check your understanding

Virtual laboratory: interference experiments

If you are interested in visualising interference phenomena, you can use our 2D Waves virtual laboratory. Below we propose a series of guided experiments, but you are free to try new configurations and new experiments. The app is very intuitive; to understand correctly what you observe, just keep in mind that the coloured areas represent the amplitude of the oscillation. The brighter the colour, the greater the amplitude of the oscillations (green for crests, red for troughs). Obviously the dark areas represent the absence of oscillation caused by destructive interference.

Suggested experiments:

• Interference between two or more point sources of the same frequency
• Interference between two point sources of different frequencies
• Interference between waves of different frequencies generated by a single source (beats)