Chapter 8
Waves
Abstract Chapter 8 deals with waves. The topics covered are wave equation, progressive and stationary waves, vibration of strings, wave velocity in solids, liquids
and gases, capillary waves and gravity waves, the Doppler effect, shock wave, reverberation in buildings, stationary waves in pipes and intensity level.
8.1 Basic Concepts and Formulae
The travelling wave: The simple harmonic progressive wave travelling in the positive x-direction can be variously written as
y = A sin
2π
λ
(vt − x)
= A sin 2π
t
T
−
x
λ
= A sin(ωt − kx)
= A sin 2π f
t −
x
v
(8.1)
where A is the amplitude, f the frequency and v the wave velocity, λ the wavelength, ω = 2π f the angular frequency and k = 2π/λ, the wave number.
Similarly, the wave in the negative x-direction can be written as
y = A sin
2π
λ
(vt + x)
(8.2)
and so on.
The superposition principle states that when two or more waves traverse the same
region independently, the displacement of any particle at a given time is given by
the vector addition of the displacement due to the individual waves.
Interference of waves: Interference is the physical effect caused by the superposition of two or more wave trains crossing the same region simultaneously. The wave
trains must have a constant phase difference.
339
Waves
Abstract Chapter 8 deals with waves. The topics covered are wave equation, progressive and stationary waves, vibration of strings, wave velocity in solids, liquids
and gases, capillary waves and gravity waves, the Doppler effect, shock wave, reverberation in buildings, stationary waves in pipes and intensity level.
8.1 Basic Concepts and Formulae
The travelling wave: The simple harmonic progressive wave travelling in the positive x-direction can be variously written as
y = A sin
2π
λ
(vt − x)
= A sin 2π
t
T
−
x
λ
= A sin(ωt − kx)
= A sin 2π f
t −
x
v
(8.1)
where A is the amplitude, f the frequency and v the wave velocity, λ the wavelength, ω = 2π f the angular frequency and k = 2π/λ, the wave number.
Similarly, the wave in the negative x-direction can be written as
y = A sin
2π
λ
(vt + x)
(8.2)
and so on.
The superposition principle states that when two or more waves traverse the same
region independently, the displacement of any particle at a given time is given by
the vector addition of the displacement due to the individual waves.
Interference of waves: Interference is the physical effect caused by the superposition of two or more wave trains crossing the same region simultaneously. The wave
trains must have a constant phase difference.
339