- Mechanical waves can exist in material media and are governed by Newton's Laws.
- Transverse waves are waves in which the particles of the medium oscillate perpendicular to the direction of wave propagation.
- Longitudinal waves are waves in which the particles of the medium oscillate along the direction of wave propagation.
- Progressive wave is a wave that moves from one point of medium to another.
The displacement in a sinusoidal wave propagating in the positive x direction is given by

where a is the amplitude of the wave, k is the angular wave number, is the angular frequency, is the phase, and φ is the phase constant or phase angle.

- Wavelength of a progressive wave is the distance between two consecutive points of the same phase at a given time. In a stationary wave, it is twice the distance between two consecutive nodes or anti nodes.
Period T of oscillation of a wave is defined as the time any element of the medium takes to move through one complete oscillation. It is related to the angular frequency

through the relation.

Frequency v of a wave is defined as

and is related to angular frequency by

The speed of a transverse wave on a stretched string is set by the properties of the string. The speed on a string with tension T and linear mass density

is

Sound waves are longitudinal mechanical waves that can travel through solids, liquids, or gases. The speed v of sound wave in a fluid having bulk modulus B and density

is

The speed of longitudinal waves in a metallic bar is

For gases, since the speed of sound is

When two or more waves traverse the same medium, the displacement of any element of the medium is the algebraic sum of the displacements due to each wave. This is known as the principle of superposition of waves

Two sinusoidal waves on the same string exhibit interference, adding or cancelling according to the principle of superposition. If the two are travelling in the same direction and have the same amplitude a and frequency but differ in phase by a phase constant

, the result is a single wave with the same frequency

:

If or an integral multiple of 2π, the waves are exactly in phase and the interference is constructive; if , they are exactly out of phase and the interference is destructive.

A travelling wave, at a rigid boundary or a closed end, is reflected with a phase reversal but the reflection at an open boundary takes place without any phase change.

For an incident wave

The reflected wave at a rigid boundary is

For reflection at an open boundary

The interference of two identical waves moving in opposite directions produces standing waves. For a string with fixed ends, the standing wave is given by

Standing waves are characterised by fixed locations of zero displacement called nodes and fixed locations of maximum displacements called antinodes. The separation between two consecutive nodes or anti nodes is

A stretched string of length L fixed at both the ends vibrates with frequencies given by

The set of frequencies given by the above relation are called the normal modes of oscillation of the system. The oscillation mode with lowest frequency is called the fundamental mode or the first harmonic. The second harmonic is the oscillation mode with n = 2 and so on.

A pipe of length L with one end closed and other end open (such as air columns)vibrates with frequencies given by

The set of frequencies represented by the above relation are the normal modes of oscillation of such a system. The lowest frequency given by is the fundamental mode or the first harmonic.

- A string of length L fixed at both ends and an air column closed at one end and open at the other end, vibrates with frequencies called its normal modes. Each of these frequencies is a resonant frequency of the system.
Beats arise when two waves having slightly different frequencies,

and comparable amplitudes, are superposed. The beat frequency is

The Doppler effect is a change in the observed frequency of a wave when the source and the observer O moves relative to the medium. For sound the observed frequency ν is given in terms of the source frequency by here v is the speed of sound through the medium, is the velocity of observer relative to the medium, and is the source velocity relative to the medium. In using this formula, velocities in the direction OS should be treated as positive and those opposite to it should be taken to be negative.

### Sample Examples

#### Question

A rocket is moving at a speed of towards a stationary target. While moving, it emits a wave of frequency 1000 Hz. Some of the sound reaching the target gets reflected back to the rocket as an echo. Calculate (1) the frequency of the sound as detected by the target and (2) the frequency of the echo as detected by the rocket.

#### Solution

Since the source is approaching a stationary target* *must be replaced by –*vs. *Thus, we have

(2) The target is now the source (because it is the source of echo) and the rocket's detector is now the detector or observer (because it detects echo). Thus,* *has a positive value. The frequency of the sound emitted by the source (the target) is , the frequency intercepted by the target and not . Therefore, the frequency as registered by the rocket is

= 4080 Hz

#### Question

A wave travelling along a string is described by, in which the numerical constants are in SI units (0.005 m, 80.0 rad m^{–1}, and3.0 rad s^{–1}). Calculate (a) the amplitude,(b) the wavelength, and (c) the period and frequency of the wave. Also, calculate the displacement y of the wave at a distance x = 30.0 cm and time t = 20 s ?

#### Solution

(a) the amplitude of the wave is 0.005 m = 5 mm.

(b) the angular wave number k and angular frequency ω are

We then relate the wavelength to k through

(c) Now we relate by the relation

and frequency,

The displacement *y *at *x = *30.0 cm and time *t = *20 s is given by