Table of Contents
Theory of waves
Traveling waves
A wave is a disturbance (usually in some medium) which travels away from its source. Some familiar examples are sound, water, and string waves. A transverse wave is a wave in which the disturbance causes a momentary displacement in the medium in a direction normal to the direction in which the wave propagates (for example, a water wave). A longitudinal wave is a compression wave in which the disturbance causes a momentary displacement of the medium along the same direction in which the wave propagates (for example, a sound wave).
Standing waves
In a one-dimensional body of finite size – such as a stretched string which is clamped at both ends – waves can travel in either direction along the body. A disturbance which travels along a string and hits a clamped end is reflected with its transverse displacement reversed from that of the incoming disturbance. If the disturbance encounters a free end of the string it will be reflected with no reversal of its transverse displacement.
The resulting motion of the body can be obtained by the superposition principle, i.e., the amplitude of the resulting wave is the sum of the individual amplitudes. For the special case when the two waves have equal amplitudes, speeds, and frequencies they may be represented by the equations
| $y_1 = y_m\sin (kx - \omega t)$ and | (1) |
| $y_2 = y_m\sin (kx + \omega t)$. |
The resulting wave therefore has an amplitude given by
| $y = y_1 + y_2 = y_m\sin (kx - \omega t) + y_m\sin (kx + \omega t)$. | (2) |
Using the trigonometric identity
| $\sin A + \sin B = 2 \sin \left(\dfrac{1}{2}(A+B)\right)\cos \left(\dfrac{1}{2}(A-B)\right)$, | (3) |
Eq. (2) may be simplified to
| $y = 2y_m\sin(kx)\cos(\omega t)$, | (4) |
which is the equation for a standing wave.
Notice that at any fixed position x, the string undergoes simple harmonic motion as time goes on. Notice also that all points along the wave oscillate at the same frequency. The amplitude, however, depends on the position x. This characteristic is quite different from a traveling wave in which the amplitudes of all points along the wave are equal.
Nodes and anti-nodes
By Eq. (4), the amplitude of a standing wave is a maximum when
$kx = \dfrac{\pi}{2},\dfrac{3\pi}{2},\dfrac{5\pi}{2},\dots$ Since $k = 2\pi / \lambda$, this corresponds to
| $x = \dfrac{\lambda}{4},\dfrac{3\lambda}{4},\dfrac{5\lambda}{4},\dots$ | (5) |
These positions of maximum amplitude are called anti-nodes.
Similarly, there are positions along the wave where the amplitude equals zero, namely where
$kx = 0,\pi ,3\pi ,\dots$ or
| $x = 0,\dfrac{\lambda}{2},\lambda ,\dfrac{3\lambda}{2},\dots$ | (6) |
These positions of zero amplitude are called nodes.
Resonant frequencies
A system like a vibrating string can be caused to vibrate with a relatively large amplitude if it is forced to oscillate at or near one of its “natural” frequencies. This phenomenon is called resonance.
For resonance to occur on a string fixed at both ends, there may be any number of nodes along the string, but the endpoints must be nodes. (See Fig. 1.) Equation (6) states that the distance between adjacent nodes is $\lambda/2$. Therefore, a string of length L vibrating at resonance must contain an integer multiple of half wavelengths,
$\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. Using the relation $f = v/\lambda$, we may re-write this as
| $f_n = \dfrac{v}{\lambda_n}=l\dfrac{nv}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (7) |
Figure 1: Standing waves on a string with both ends fixed.
For a string with one end fixed and the other end free (or for sound waves in an air column with one end open and one end closed), the fixed end must be a node and resonances will occur when the free end is an anti-node. (See Fig. 2.) Here, the string (or air column) length must contain an odd integer multiplier of quarter-wave lengths,
$\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$ such that the resonant frequencies will be
| $f_n = \dfrac{v}{\lambda_n} = \dfrac{nv}{4L} \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$. | (8) |
Figure 2: Standing waves on a string with one free and one fixed end. Hover for animation
If both ends of the string are free, resonance will occur when there are anti-nodes at both ends. (See Fig. 3.) At resonance, the string length will contain integer multiples of half wavelengths,
$\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$ with resonant frequencies
| $f_n = \dfrac{v}{\lambda_n}=\dfrac{nv}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (9) |
This is the same result as Eq. (7) for a string with both ends fixed.
Figure 3: Standing waves on a string with both ends free. Hover for animation
Wave propagation speed
The speed of propagation of a wave is given by a formula of the form
| $v = \sqrt{\dfrac{\textrm{“restoring force” factor}}{\textrm{“inertial” factor}}}$. | (10) |
For a transverse wave along a string, Eq. (10) is more explicitly
| $v_{string} = \sqrt{\dfrac{T}{\rho}}$, | (11) |
where T is the tension in the string and $\rho$ (rho) is the mass per unit length of the stretched string.
For a stretched spring, Eq. (10) becomes
| $v_{spring} = \sqrt{\dfrac{kL}{\rho}}$, | (12) |
where k is the spring constant, L is the length of the stretched spring and $\rho$ is the mass per unit length of the stretched spring. The speed of propagation of a sound wave is
| $V_{gas} = \sqrt{\dfrac{\gamma k_BT}{m}}$, | (13) |
where $\gamma$ is the ratio of specific heats of the gas in which the wave is moving ( for a diatomic gas), _k_B is the Boltzmann constant, T is the absolute temperature of the gas (in Kelvin), and m is the mass of an individual molecule of gas. (The mass of an air molecule is about 4.8 x 10-26 kg.) We may use Eqs. (11)-(13) to express the predicted frequencies in terms of variables we can control and measure in the laboratory. Substituting Eq. (11) into Eq. (7) (which is the same as Eq. (9)), gives the resonant frequencies for a string, fixed at both ends or free at both ends as
| $f_n = \dfrac{n}{2L}\sqrt{\dfrac{T}{\rho}} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$, | (14) |
and for a spring, fixed at both ends or free at both ends as
| $f_n = \dfrac{n}{2L}\sqrt{\dfrac{kL}{\rho}} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (15) |
Similarly, substituting Eq. (13) into Eq. (8) gives the resonant frequencies for an air column with one end closed and the other end open as
| $f_n = \dfrac{n}{4L}\sqrt{\dfrac{\gamma k_BT}{m}} \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$. | (16) |
Substituting Eq. (13) into Eq. (9) gives the resonant frequencies for an air column with both ends open as
| $f_n = \dfrac{n}{2L}\sqrt{\dfrac{\gamma k_BT}{m}} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (17) |