Vibrating stringsIf a string stretched between two points is plucked it vibrates, and a wave travels along the string. Since the vibrations are from side to side the wave is transverse. The velocity of the wave along the string can be found as follows.

Velocity of waves along a stretched string

Assume that the velocity of the wave v depends upon(a) the tension in the string (T),(b) the mass of the string (M) and(c) the length of the string (L) (see Figure 1).

Therefore: v = kTxMyLz

Solving this gives x = , z = , y = - .The constant k can be shown to be equal to 1 in this case and we write m as the mass per unit length where m = M/L. The formula therefore becomes:

velocity of waves on a stretched string = [T/m]1/2

Since velocity = frequency x wavelength

Frequency of a vibrating string = [T/m]1/2

The Physics of vibrating stringsA string is fixed between two points. If the centre of the string is plucked vibrations move out in opposite directions along the string. This causes a transverse wave to travel along the string. The pulses travel outwards along the string and when they reaches each end of the string they are reflected (see Figure 2).

The two travelling waves then interfere with each other to produce a standing wave in the string. In the fundamental mode of vibration there are points of no vibration or nodes at each end of the string and a point of maximum vibration or antinode at the centre.

Notice that there is a phase change when the pulse reflects at each end of the string.

The first three harmonics for a vibrating string are shown in the following diagrams.

(a) As has already been shown; for a string of length L and mass per unit length m under a tension T the fundamental frequency is given by:

Frequency (f) = 1/2L[T/m]1/2

(b) First overtone or second harmonic:

Frequency (f) = 1/L[T/m]1/2

(c) Second overtone or third harmonic:

Frequency (f) = 3/2L[T/m]1/2

A string can be made to vibrate in a selected harmonic by plucking it at one point (the antinode) to give a large initial amplitude and touching it at another (the node) to prevent vibration at that point.

The DopplerEffect

Most everyone is familiar with the drop in pitch of a train whistle as a train passes your position and switches from moving toward you to moving away from you. This phenomenon is called theDoppler Effect, and is associated with the wave nature of sound: the relative motion of the source causes the wavelength of the sound waves to be decreased ahead of the source and stretched out behind the source (musically, the pitch of a note is correlated with the wavelength of the corresponding sound wave; thus, the longer the wavelength, the lower the pitch). Here is aJava Virtual Experimentthat illustrates the Doppler effect.Light also can be described as a wave, and relative motion of the source of light waves leads to a corresponding Doppler effect for light. In this case it is not the pitch but the color (that is, the wavelength) that is shifted by the motion of the source. The wavelength is shifted to larger values if the motion of the source is away from the observer and to smaller values if the motion is toward the observer. The shift to larger wavelengths by motion away from the observer is called ared shiftby astronomers and a shift to shorter wavelengths caused by motion toward the observer is called ablue shift. The terminology is borrowed from the visible part of the spectrum where blue is toward the short wavelength end and red is toward the long wavelength end, but the Doppler effect occurs for all wavelengths of light, not just the visible spectrum.

Resonance- when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion. Resonance is a common cause of sound production in musical instruments. One of our best models of resonance in a musical instrument is a resonance tube (ahollow cylindrical tube) partially filled with water and forced into vibration by a tuning fork. The tuning fork is the object that forced the air inside of the resonance tube into resonance. As the tines of the tuning fork vibrate at their own natural frequency, they created sound waves that impinge upon the opening of the resonance tube. These impinging sound waves produced by the tuning fork force air inside of the resonance tube to vibrate at the same frequency. Yet, in the absence of resonance, the sound of these vibrations is not loud enough to discern. Resonance only occurs when the first object is vibrating at the natural frequency of the second object. So if the frequency at which the tuning fork vibrates is not identical to one of the natural frequencies of the air column inside the resonance tube, resonance will not occur and the two objects will not sound out together with a loud sound. But the location of the water level can be altered by raising and lowering a reservoir of water, thus decreasing or increasing the length of the air column.Resonance and Musical InstrumentsMusical instruments produce their selected sounds in the same manner. Brass instruments typically consist of a mouthpiece attached to a long tube filled with air. The tube is often curled in order to reduce the size of the instrument. The metal tube merely serves as a container for a column of air. It is the vibrations of this column that produces the sounds that we hear. The length of the vibrating air column inside the tube can be adjusted either by sliding the tube to increase and decrease its length or by opening and closing holes located along the tube in order to control where the air enters and exits the tube. Brass instruments involve the blowing of air into a mouthpiece. The vibrations of the lips against the mouthpiece produce a range of frequencies. One of the frequencies in the range of frequencies matches one of the natural frequencies of the air column inside of the brass instrument. This forces the air inside of the column into resonance vibrations.The result of resonance is always a big vibration - that is, a loud sound.Woodwind instruments operate in a similar manner. Only, the source of vibrations is not the lips of the musician against a mouthpiece, but rather the vibration of a reed or wooden strip.The operation of a woodwind instrument is often modeled in a Physics class using a plastic straw. The ends of the straw are cut with a scissors, forming a taperedreed. When air is blown through the reed, the reed vibrates producing turbulence with a range of vibrational frequencies. When the frequency of vibration of the reed matches the frequency of vibration of the air column in the straw, resonance occurs. And once more, the result of resonance is a big vibration - the reed and air column sound out together to produce a loud sound. As if this weren't silly enough, the length of the straw is typically shortened by cutting small pieces off its opposite end. As the straw (and the air column that it contained) is shortened, the wavelength decreases and the frequency was increases. Higher and higher pitches are observed as the straw is shortened. Woodwind instruments produce their sounds in a manner similar to the straw demonstration. A vibrating reed forces an air column to vibrate at one of its natural frequencies. Only for wind instruments, the length of the air column is controlled by opening and closing holes within the metal tube (since the tubes are a little difficult to cut and a too expensive to replace every time they are cut).Resonance is the cause of sound production in musical instruments.