Standing waves occur when two waves of the same frequency and amplitude move in opposite directions and interfere with each other. This process of wave addition at every point is called interference.
The rightward traveling wave can be described as:
$$
y_1(x,t) = A \sin(kx - \omega t)
$$
The leftward traveling wave is:
$$
y_2(x,t) = A \sin(kx + \omega t)
$$
The total displacement is the sum of the two:
$$
y(x,t) = y_1(x,t) + y_2(x,t) = A[\sin(kx - \omega t) + \sin(kx + \omega t)]
$$
Using the trigonometric identity:
$$
\sin a + \sin b = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right)
$$
we get the result:
$$
\boxed{y(x,t) = 2A \sin(kx) \cos(\omega t)}
$$
This describes a wave that depends on both time and space but does not propagate — it's a stationary oscillation at every position $x$. Such a wave is called a standing wave.
In this pattern, there are points with maximum oscillation amplitude called antinodes, and points where no oscillation occurs, called nodes. The distance between two adjacent nodes or antinodes is always $\lambda/2$, so the distance between a node and an adjacent antinode is $\lambda/4$.
In wind instruments, standing waves are formed inside pipes. These instruments often have openings at both ends, where antinodes must occur. The length $l$ of the pipe must satisfy:
$$
\boxed{l = (n + 1)\frac{\lambda}{2}}
$$
The case $n = 0$ corresponds to the fundamental frequency, and higher $n$ correspond to overtones or harmonics.
If one end of the pipe is closed (a stopped pipe), there must be a node at the closed end and an antinode at the open end. In this case:
$$
\boxed{l = (2n - 1)\frac{\lambda}{4}}
$$
Some examples of fundamental and overtone modes in both open and closed pipes are shown in the following figure:
Timbre of musical instruments:
Pure sine tones (single-frequency sounds) are rare in nature. Most sounds consist of a mixture of several harmonics of different amplitudes. This gives each instrument or voice its unique timbre. That's why a violin and a piano sound different even if they play the same note.
In string instruments:
Standing waves also occur in stringed instruments like the guitar. The string has fixed nodes at both ends, and the wave forms antinodes in between depending on the harmonic.
In microwave ovens:
Standing waves can cause heating problems in microwaves. The reflected electromagnetic waves form stationary patterns with high and low intensity regions. This is why microwaves use rotating plates to ensure even heating.
Example problem:
A closed organ pipe should produce the fundamental tone of A = 440 Hz.
If the speed of sound is assumed to be 330 m/s, what length must the pipe have?
Solution:
$$
l = \frac{\lambda}{4} = \frac{c}{4f} = \frac{330}{4 \cdot 440} \approx 0.1875\,\text{m} = 18.75\,\text{cm}
$$
Experiments
1. Standing Microwaves
If you remove the rotating plate from a microwave and place a chocolate bar inside, melted spots will appear at the antinodes of the standing wave. Measure the distance between these spots, multiply by 2 to get the wavelength, and use the microwave’s printed frequency to calculate the speed of light:
$$
c = f \cdot \lambda
$$
2. Kundt’s Tube Experiment
A transparent horizontal tube contains cork powder and has one open and one closed end. A speaker generates a sound wave on the open side, and the length of the air column can be varied using a plunger. When a standing wave forms, the cork collects at the nodes. Measuring the spacing between these nodes allows for accurate determination of the speed of sound in air or other gases.