348
8 Waves
(b) The wave function of a standing wave on a string that is fixed at both ends
is given in SI units by y(x, t) = (0.024) sin(62.8x) cos(471t).
Find the speed of the waves on the string, and the distance between nodes
for the standing wave.
[hint: You may need to use sin θ 1 + sin θ 2 = 2 cos
1
2 (θ 2 − θ 1 ) sin
1
2 (θ 1 +
θ 2 )].
8.24 A progressive wave travelling along a string has maximum amplitude
A = 0.0821 m, angular frequency ω = 100 rad/s and wave number k =
22.0 rad/m. If the wave has zero amplitude at t = 0 and x = 0 for its starting
conditions
(i) State the wave function that represents the progressive wave motion for
this wave travelling in the negative x-direction.
(ii) State the wave function for this wave travelling in the positive
x-direction.
(iii) Find the wavelength (λ), period (T ) and the speed (v) of this wave.
(iv) Find its amplitude at a time t = 2.5 s at a distance x = 3.2 m from its
origin, for this wave travelling in the negative x-direction.
[University of Wales 2008]
8.25 The speed of a wave on a string is given by v =
F
μ . Show that the right-hand
side of this equation has the units of speed.
8.26 For a sinusoidal wave travelling along a string show that at any time t the
slope
∂ y
∂ x at any point x is equal to the negative of the instantaneous transverse
velocity
∂ y
∂t of the string at x divided by the wave velocity v.
8.27 (a) Consider a small segment of a string upon which a wave pulse is travelling.
Using this diagram, or otherwise, show that the wave equation for transverse waves on a stretched string is
∂ 2 y
∂ x 2 =
μ
F
∂ 2 y
∂t 2
where μ is the mass per unit length and F is the tension.
(b) Show that the wave function representing a wave travelling in the positive x-direction, y(x − vt), is a solution of the wave equation. Obtain an
expression for the velocity, v, of the wave (Fig. 8.1).
8.28 Two wires of different densities are joined as in Fig. 8.2. An incident wave
y 1 = A 1 sin(ωt − k 1 x) travelling in the positive x-direction along the wire
at the boundary is partly transmitted. (a) Find the reflected and transmitted
amplitudes in terms of the incident amplitude. (b) When will the amplitude of
the reflected wave be negative?
8 Waves
(b) The wave function of a standing wave on a string that is fixed at both ends
is given in SI units by y(x, t) = (0.024) sin(62.8x) cos(471t).
Find the speed of the waves on the string, and the distance between nodes
for the standing wave.
[hint: You may need to use sin θ 1 + sin θ 2 = 2 cos
1
2 (θ 2 − θ 1 ) sin
1
2 (θ 1 +
θ 2 )].
8.24 A progressive wave travelling along a string has maximum amplitude
A = 0.0821 m, angular frequency ω = 100 rad/s and wave number k =
22.0 rad/m. If the wave has zero amplitude at t = 0 and x = 0 for its starting
conditions
(i) State the wave function that represents the progressive wave motion for
this wave travelling in the negative x-direction.
(ii) State the wave function for this wave travelling in the positive
x-direction.
(iii) Find the wavelength (λ), period (T ) and the speed (v) of this wave.
(iv) Find its amplitude at a time t = 2.5 s at a distance x = 3.2 m from its
origin, for this wave travelling in the negative x-direction.
[University of Wales 2008]
8.25 The speed of a wave on a string is given by v =
F
μ . Show that the right-hand
side of this equation has the units of speed.
8.26 For a sinusoidal wave travelling along a string show that at any time t the
slope
∂ y
∂ x at any point x is equal to the negative of the instantaneous transverse
velocity
∂ y
∂t of the string at x divided by the wave velocity v.
8.27 (a) Consider a small segment of a string upon which a wave pulse is travelling.
Using this diagram, or otherwise, show that the wave equation for transverse waves on a stretched string is
∂ 2 y
∂ x 2 =
μ
F
∂ 2 y
∂t 2
where μ is the mass per unit length and F is the tension.
(b) Show that the wave function representing a wave travelling in the positive x-direction, y(x − vt), is a solution of the wave equation. Obtain an
expression for the velocity, v, of the wave (Fig. 8.1).
8.28 Two wires of different densities are joined as in Fig. 8.2. An incident wave
y 1 = A 1 sin(ωt − k 1 x) travelling in the positive x-direction along the wire
at the boundary is partly transmitted. (a) Find the reflected and transmitted
amplitudes in terms of the incident amplitude. (b) When will the amplitude of
the reflected wave be negative?