Question

Wave motion is characterized by two velocities: the velocity with which the wave moves in the medium (e.g., air or a string) and the velocity of the medium (the air or the string itself). Consider a transverse wave traveling in a string. The mathematical form of the wave is y(x,t)=Asin(kx−ωt).
a)Find the velocity of propagation v_p of this wave.

Express the velocity of propagation in terms of some or all of the variables A, k, and omega.

b)Find the y velocity v_y(x,t) of a point on the string as a function of x and t.

Express the y velocity in terms of omega, A, k, x, and t.

Answer 1

Answer: a) v = ω /k, b) v = - ωAcos( kx −ωt)

Explanation:

y(x,t)=Asin(kx−ωt) defines the wave equation.

a)

We are asked to find wave speed (v)

Recall that v = fλ

From the wave equation above,

k = 2π/ λ where k is the wave number and λ is the wavelength, λ = 2π /k

ω = 2πf where f is the frequency and ω is the angular frequency.

f = ω/ 2π.

By substituting for λ and ω into the wave speed formulae, we have that

v =( ω/ 2π) × (2π /k)

v = ω/k

b)

y(x,t)=Asin(kx−ωt)

The first derivative of y with respect to x give the velocity (vy)

By using chain rule, we have that

v = dy/dt = A cos( kx −ωt) × (−ω)

v = - ωAcos( kx −ωt)