350
8 Waves
8.31 (a) Show that when a string of length L plucked at the centre through height
h, the energy in the nth mode is given by E n =
16Mh 2 v 2
n 2 π 2 L 2 , where v is the
wave velocity and M is the total mass of the string.
(b) Compare the energies in the first and the third harmonics of a string
plucked at the centre.
8.2.2 Waves in Solids
8.32 (a) A steel bar of density 7860 kg/m
3 and Young’s modulus 2 × 10 11 N/m 2
and of length 0.25 m is rigidly clamped at one end and free to move at the
other end. Determine the fundamental frequency of the bar for longitudinal harmonic vibrations.
(b) How do the frequencies compare with (i) rod free at both ends; (ii) bar
clamped at the midpoint; and (iii) bar clamped at both the ends.
8.33 A 2 kg mass is hung on a steel wire of 1 × 10 −5 m 2 cross-sectional area and
1.0 m length. (a) Calculate the fundamental frequency of vertical oscillations
of the mass by considering it to be a simple oscillator and (b) calculate the
fundamental frequency of vertical oscillations of the mass by regarding it as a
system of longitudinally vibrating bar fixed at one end and mass-loaded at the
other. Assume Y = 210 11 N/m 2 and ρ = 7800 kg/m
2 for steel.
8.34 Show that for kl < 0.2, the frequency equation derived for the mass loaded
system for the bar of length l clamped at one end and loaded at the other
reduces to that of a simple harmonic oscillator (you may assume that the frequency condition for this system is, kl tan kl = M/m).
8.2.3 Waves in Liquids
8.35 (a) Find the velocity of long waves for a liquid whose depth is λ/4 and compare it with (b) the velocity for a similar wavelength λ in a deep liquid and (c)
that for canal waves.
8.36 Find the maximum depth of liquid for which the formula v 2 = gh represents
the velocity of waves of length λ within 1%. You may assume that the velocity
of surface waves is given by v =
g tanh(kh)
k
which is valid for relatively deep
waters.
8.37 In an experiment to measure the surface tension of water by the ripple method,
the waves were created by a tuning fork of frequency 100 Hz and the wavelength was 3.66 mm. Calculate the surface tension of water.
8 Waves
8.31 (a) Show that when a string of length L plucked at the centre through height
h, the energy in the nth mode is given by E n =
16Mh 2 v 2
n 2 π 2 L 2 , where v is the
wave velocity and M is the total mass of the string.
(b) Compare the energies in the first and the third harmonics of a string
plucked at the centre.
8.2.2 Waves in Solids
8.32 (a) A steel bar of density 7860 kg/m
3 and Young’s modulus 2 × 10 11 N/m 2
and of length 0.25 m is rigidly clamped at one end and free to move at the
other end. Determine the fundamental frequency of the bar for longitudinal harmonic vibrations.
(b) How do the frequencies compare with (i) rod free at both ends; (ii) bar
clamped at the midpoint; and (iii) bar clamped at both the ends.
8.33 A 2 kg mass is hung on a steel wire of 1 × 10 −5 m 2 cross-sectional area and
1.0 m length. (a) Calculate the fundamental frequency of vertical oscillations
of the mass by considering it to be a simple oscillator and (b) calculate the
fundamental frequency of vertical oscillations of the mass by regarding it as a
system of longitudinally vibrating bar fixed at one end and mass-loaded at the
other. Assume Y = 210 11 N/m 2 and ρ = 7800 kg/m
2 for steel.
8.34 Show that for kl < 0.2, the frequency equation derived for the mass loaded
system for the bar of length l clamped at one end and loaded at the other
reduces to that of a simple harmonic oscillator (you may assume that the frequency condition for this system is, kl tan kl = M/m).
8.2.3 Waves in Liquids
8.35 (a) Find the velocity of long waves for a liquid whose depth is λ/4 and compare it with (b) the velocity for a similar wavelength λ in a deep liquid and (c)
that for canal waves.
8.36 Find the maximum depth of liquid for which the formula v 2 = gh represents
the velocity of waves of length λ within 1%. You may assume that the velocity
of surface waves is given by v =
g tanh(kh)
k
which is valid for relatively deep
waters.
8.37 In an experiment to measure the surface tension of water by the ripple method,
the waves were created by a tuning fork of frequency 100 Hz and the wavelength was 3.66 mm. Calculate the surface tension of water.