Chapter 15b Homework

Particle velocity versus Wave velocity

Qu. 1 A certain transverse wave is described by

y(x,t) = A Cos [2π ( x – vt)/λ]

a) Use y(x,t) to find an expression for the transverse velocity, vy, of a

particle in the string on which the wave travels.

(Ans: δy/δt = (2πAv/λ) Sin [2π ( x – vt)/λ] )

b) Find the maximum velocity of a particle in the string

(Ans: [δy/δt]max = 2πAv/λ )

c) Under what circumstances is the maximum particle velocity equal to

the wave velocity v? (Ans: [δy/δt]max = v when λ = 2πA )

d) Under what circumstances is the maximum particle velocity less than

the wave velocity v? (Ans: [δy/δt]max < v when λ > 2πA )

e) Under what circumstances is the maximum particle velocity greater

than the wave velocity v? (Ans: [δy/δt]max > v when λ < 2πA )

Waves on a String

Qu. 2 A simple harmonic oscillator at the point x=0 generates a wave on a

string. The oscillator operates at a frequency of 40 Hz and an amplitude of 3

cm. The rope has a linear mass density of 50 g/m and is stretched with a

tension of 5 N.

a) Determine the speed of the wave (Ans: v = 10 m/s )

b) Find the wavelength (Ans: λ = 0.25 m)

c) Write the wave function y(x,t) for the wave. Assume that the

oscillator has it’s maximum upward displacement at t= 0.

(Ans: y(x,t) = 0.03 Cos [2π ( x – 10 t)/ 0.25]

d) Find the maximum transverse acceleration of points on the string.

(Ans: 1895 m/s2 )

e) Given your answer to part d) is it reasonable to ignore the effects of

gravity? (Ans: Yes !, since [δ2y/δt

2]max >> g )