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8 Waves
8.5 A sinusoidal wave on a string travelling in the +x direction at 8 m/s has a
wavelength 2 m. (a) Find its wave number, frequency and angular frequency.
(b) If the amplitude is 0.2 m, and the point x = 0 on the string is at its equilibrium position (y = 0) at time t = 0, find the equation for the wave.
8.6 A sinusoidal wave on a string travelling in the +x direction has wave number
3/m and angular frequency 20 rad/s. If the amplitude is 0.2 m and the point
at x = 0 is at its maximum displacement and t = 0, find the equation of the
wave.
8.7 Show that when a standing wave is formed, each point on the string is
undergoing SHM transverse to the string.
8.8 The length of the longest string in a piano is 2.0 m and the wave velocity of
the string is 120 m/s. Find the frequencies of the first three harmonics.
8.9 Two strings are tuned to fundamentals of f 1 = 4800 Hz and f
1 = 32 Hz. Their
lengths are 0.05 and 2.0 m, respectively. If the tension in these two strings is
the same, find the ratio of the masses per unit length of the two strings.
8.10 The equation of a transverse wave travelling on a rope is given by y =
5 sin π(0.02x − 4.00t), where y and x are expressed in centimetres and t is in
seconds. Find the amplitude, frequency, velocity and wavelength of the wave.
8.11 A string vibrates according to the equation y = 4 sin
1
2 π x cos 20π t, where
x and y are in centimetres and t is in seconds. (a) What are the amplitudes
and velocity of the component waves whose superposition can give rise to this
vibration? (b) What is the distance between the nodes? (c) What is the velocity
of the particle in the transverse direction at x = 1.0 cm and when t = 9/4 s?
8.12 A wave of frequency 250 cycles/s has a phase velocity 375 m/s. (a) How
far apart are two points 60 ◦ out of phase? (b) What is the phase difference
between two displacements at a certain point at time 10 −3 s apart?
8.13 Two sinusoidal waves having the same frequency and travelling in the same
direction are combined. If their amplitudes are 6.0 and 8.0 cm and have a phase
difference of π/2 rad, determine the amplitude of the resultant motion.
8.14 Show that the one-dimensional wave equation is satisfied by the following
functions:
(a) y = A ln(x + vt) and (b) y = A cos(x + vt).
8.15 (a) A cord of length L is rigidly attached at both ends and is plucked to a
height h at a point 1/3 from one end and let it go. Show that the displacement y at any distance x along the string at time t in the subsequent
motion is given by
y =
3 5/2
2π 2
sin
π x
L
cos
πvt
L
+
1
4
sin
2π x
L
cos
2πvt
L
−
1
16
sin
4π x
L
cos
4πvt
L
. . .
(b) and that the third, sixth and ninth harmonics are absent.
8 Waves
8.5 A sinusoidal wave on a string travelling in the +x direction at 8 m/s has a
wavelength 2 m. (a) Find its wave number, frequency and angular frequency.
(b) If the amplitude is 0.2 m, and the point x = 0 on the string is at its equilibrium position (y = 0) at time t = 0, find the equation for the wave.
8.6 A sinusoidal wave on a string travelling in the +x direction has wave number
3/m and angular frequency 20 rad/s. If the amplitude is 0.2 m and the point
at x = 0 is at its maximum displacement and t = 0, find the equation of the
wave.
8.7 Show that when a standing wave is formed, each point on the string is
undergoing SHM transverse to the string.
8.8 The length of the longest string in a piano is 2.0 m and the wave velocity of
the string is 120 m/s. Find the frequencies of the first three harmonics.
8.9 Two strings are tuned to fundamentals of f 1 = 4800 Hz and f
1 = 32 Hz. Their
lengths are 0.05 and 2.0 m, respectively. If the tension in these two strings is
the same, find the ratio of the masses per unit length of the two strings.
8.10 The equation of a transverse wave travelling on a rope is given by y =
5 sin π(0.02x − 4.00t), where y and x are expressed in centimetres and t is in
seconds. Find the amplitude, frequency, velocity and wavelength of the wave.
8.11 A string vibrates according to the equation y = 4 sin
1
2 π x cos 20π t, where
x and y are in centimetres and t is in seconds. (a) What are the amplitudes
and velocity of the component waves whose superposition can give rise to this
vibration? (b) What is the distance between the nodes? (c) What is the velocity
of the particle in the transverse direction at x = 1.0 cm and when t = 9/4 s?
8.12 A wave of frequency 250 cycles/s has a phase velocity 375 m/s. (a) How
far apart are two points 60 ◦ out of phase? (b) What is the phase difference
between two displacements at a certain point at time 10 −3 s apart?
8.13 Two sinusoidal waves having the same frequency and travelling in the same
direction are combined. If their amplitudes are 6.0 and 8.0 cm and have a phase
difference of π/2 rad, determine the amplitude of the resultant motion.
8.14 Show that the one-dimensional wave equation is satisfied by the following
functions:
(a) y = A ln(x + vt) and (b) y = A cos(x + vt).
8.15 (a) A cord of length L is rigidly attached at both ends and is plucked to a
height h at a point 1/3 from one end and let it go. Show that the displacement y at any distance x along the string at time t in the subsequent
motion is given by
y =
3 5/2
2π 2
sin
π x
L
cos
πvt
L
+
1
4
sin
2π x
L
cos
2πvt
L
−
1
16
sin
4π x
L
cos
4πvt
L
. . .
(b) and that the third, sixth and ninth harmonics are absent.