8.2 Problems
345
where V is in cubic metres and S in square metre for the volume and surface area
of the room, respectively, and K is the absorption coefficient of the material of the
floor, ceiling, walls, etc. summed over these components.
Beats: When two wave trains of slightly different frequencies travel through the
same region, a regular swelling and fading of the sound is heard, a phenomenon
called beats.
At a given point let the displacements produced by the two waves be
y = A sin ω 1 t
(8.36)
y = A sin ω 2 t
(8.37)
By the superposition principle, the resultant displacement is given by
y = y 1 + y 2 = [2A cos 2π( f 1 − f 2 )t/2] sin 2π( f 1 + f 2 )t/2
(8.38)
The resulting vibration has a frequency
f = ( f 1 + f 2 )/2
(8.39)
and an amplitude given by the expression in the square bracket of (8.38). The beat
frequency is given by f 1 ∼ f 2 .
8.2 Problems
8.2.1 Vibrating Strings
8.1 Show that the one-dimensional wave equation is satisfied by the function
y = A
√ (x + vt).
8.2 Show that the equation y = 2A sin(nπx/L) cos 2π f t for a standing wave is a
solution of the wave equation
∂ 2 y
∂ x 2 =
μ
F
∂ 2 y
∂t 2
where F is the tension and μ the mass/unit length.
8.3 A cord of length L fixed at both ends is set in vibration by raising its centre a
distance h and let go. Obtain an expression for the displacement y at any point
x and time t as a series expansion assuming that initially the velocity is zero.
Also show that even harmonics are absent.
8.4 Show that the superposition of the waves y 1 = A sin(kx − ωt) and y 2 =
3A sin(kx + ωt) is a pure standing wave plus a travelling wave in the negative
direction along the x-axis. Find the amplitude of (a) the standing wave and (b)
the travelling wave.
345
where V is in cubic metres and S in square metre for the volume and surface area
of the room, respectively, and K is the absorption coefficient of the material of the
floor, ceiling, walls, etc. summed over these components.
Beats: When two wave trains of slightly different frequencies travel through the
same region, a regular swelling and fading of the sound is heard, a phenomenon
called beats.
At a given point let the displacements produced by the two waves be
y = A sin ω 1 t
(8.36)
y = A sin ω 2 t
(8.37)
By the superposition principle, the resultant displacement is given by
y = y 1 + y 2 = [2A cos 2π( f 1 − f 2 )t/2] sin 2π( f 1 + f 2 )t/2
(8.38)
The resulting vibration has a frequency
f = ( f 1 + f 2 )/2
(8.39)
and an amplitude given by the expression in the square bracket of (8.38). The beat
frequency is given by f 1 ∼ f 2 .
8.2 Problems
8.2.1 Vibrating Strings
8.1 Show that the one-dimensional wave equation is satisfied by the function
y = A
√ (x + vt).
8.2 Show that the equation y = 2A sin(nπx/L) cos 2π f t for a standing wave is a
solution of the wave equation
∂ 2 y
∂ x 2 =
μ
F
∂ 2 y
∂t 2
where F is the tension and μ the mass/unit length.
8.3 A cord of length L fixed at both ends is set in vibration by raising its centre a
distance h and let go. Obtain an expression for the displacement y at any point
x and time t as a series expansion assuming that initially the velocity is zero.
Also show that even harmonics are absent.
8.4 Show that the superposition of the waves y 1 = A sin(kx − ωt) and y 2 =
3A sin(kx + ωt) is a pure standing wave plus a travelling wave in the negative
direction along the x-axis. Find the amplitude of (a) the standing wave and (b)
the travelling wave.