Waves

A wave is, in general, a disturbance that moves through a medium.

Examples:

wave on a string,

sound wave,

electromagnetic wave (light) can travel through a vacuum.

In all cases the wave carries energy but there is no transport of matter.

Mathematical representation of travelling waves

The wave displacement is a function of the position, x, and the time, t

x = f(x,t)

For the wave moving in the positive x direction with a constant speed c the displacement is represented by

x = g(ct — x)

(For the wave moving in the negative x direction y = g(ct + x)

The most common form of a travelling periodic wave is called a harmonic wave, or a sinusoidal wave. In mathematical representation of a harmonic wave

Amplitude (A) is the maximum displacement of any particle from its equilibrium position.

The distance between two successive portions on the wave having the same displacement at the same phase (angle) is called a wavelength (l) of the wave.

The time to go l is a period of the wave, T.

The frequency, f, measured in Hertz (Hz), is the number of repeats that pass a given point per unit time.

f = 1/T (1Hz = 1s-1)

Since the distance l is travelled in time T, the wave speed can be calculated as the ratio of l and T,

c = l/T

c is also called the phase speed.

More convenient form of mathematical representation of displacement:

The constant

is called the wave number (in m-1), and

is the angular frequency (in rad-1).

An argument of sin function

is called the phase of the wave.

In general

Where

is called a phase constant.

Types of wave motion

In a longitudinal wave the displacement is in the direction of wave propagation.

In a transverse wave the displacement is perpendicular to the direction of propagation.

Waves interaction

Constructive or destructive interference

The superposition principle applies if two or more waves pass through the same medium simultaneously. Then the resultant wave’s displacement at any point x will be the addition of the displacement of the component waves.

x(x,t) = x1(x,t) + x2(x,t)

Where the interaction of the waves leads to cancellation or reinforcement it is called destructive or constructive interference respectively.

Consider two waves travelling in the same direction, with the same amplitude, frequency and speed :

x1(x,t) = Asin(wt- kx)

x2(x,t) = Asin(wt- kx +fo)

The combined wave is:

x(x,t) = A[sin(wt- kx) + sin(wt- kx+fo)]

Applying

gives

If fo= 0 then x(x,t) = 2Asin(wt - kx)

If fo= p(or 180o) then x(x,t) = 0

For x > A, constructiveinterference.

And for x < A, destructive interference.

 

Standing wave

For two waves travelling in the opposite direction, with the same amplitude, frequency and speed, the standing wave is produced.

For the wave travelling to the right

xR = Asin(wt - kx)

For the wave travelling to the left

xL = Asin(wt + kx)

The resultant standing wave is

x = xR + xL = Asin(wt - kx) + Asin(wt + kx)

= 2Asin(kx)cos(wt)

Any particle will vibrate with SHM due to the cos factor and all will have the same frequency f = w/2p, but the amplitude depends on x and equals 2Asin(kx).

The maximum amplitude or antinode occurs when x = 2A , that is when sin(kx) = 1 that is when x = p/2k, 3p/2k, 5p/2k, … or x = l/4, 3l/4, 5l/4, …

In general antinodes for displacement (or nodes for pressure) are observed when

x = (2n+1)l/4 where n = 0, 1, 2, 3,…

Zero displacement or node occurs when

x = 0, that is when sin(kx) = 0.

So nodes occur when kx = 0, p, 2p, 3p, … or when x = 0, l/2, l, 3l/2, etc

In general nodes for displacement (or antinodes for pressure) are formed when

x = nl/2 where n = 0, 1, 2, 3, …

No energy is transferred in a standing wave as no energy passes through a node. The particles in the medium just execute SHM up and down. The distance between nodes and antinodes is half a wavelength, l/2.

Standing waves in open tube (open at both ends)

As there are antinodes at both ends, the tube can maintain all of the harmonics as its overtones. The wavelength is given by

ln = 2L/n

where L is a length of tube and n changes as 1, 2, 3, …

The frequency of harmonics is given by

fn = nc/2L

Standing waves in closed tube (open at one end)

As there must be a node-antinode pair, the harmonics are odd multiples of the fundamental only

fm = mc/4L (m = 1,3,5, …)

Note that fundamental frequency for a closed tube is half that for an open tube.

 

The speed of a wave in a medium is related to the elastic and inertial properties of that medium by

 

 

For a stretched string

where T is the tension in the string and m is the mass per unit length.

For air or water

where B represents the bulk modulus of a liquid or gas and r is a density of medium.

For solid medium

Where E is the modulus of elasticity and r is a density.

Sound Waves

Sound waves are longitudinal vibrations of material media - solid, liquid or gaseous.

Matter must be present - no sound propagates in a vacuum!

Sources of sound wave are usually mechanical vibrations caused by impacts, friction and other forms of energy dissipation.

A sound wave with amplitude A, velocity c and wavelength l is travelling such that, the displacement x from the mean position at time t and position x is given by the usual equation for a wave

where r is position in polar co-orinates

 

The velocity of wave

Also

Pressure

The average total energy passing through unit area in unit time is the intensity

Since amplitude of the displacement is related to the pressure amplitude, Pm

It is easier to measure pressure due to a wave rather than the amplitude of the displacement!

Intensities in acoustics range from 1,000,000 to 1.

The logarithmic scale (a compressed scale) known as the decibel scale is used to defined the intensity level

Where I is the intensity and Io is the reference intensity (10-12Wm-2).

Since intensity is proportional to the square of pressure due to wave, in terms of the average r.m.s.(root mean squared) pressure due to wave