Question

The equation of a plane standing wave in a homogeneous elastic medium has the form ℰ = a cos kx• cos ωt. Plot: 

(a) ℰ and ∂ℰ/∂x as functions of x at the moments t = 0 and t = T/2, where T is the oscillation period; 

(b) the distribution of density ρ (x) of the medium at the moments t = 0 and t = T/2 in the case of longitudinal oscillations; 

(c) the velocity distribution of particles of the medium at the moment t = T/4; indicate the directions of velocities at the antinodes, both for longitudinal and transverse oscillations.

Answer 1

We are given

(a) The graphs of  are as shown in Fig. (35) of the book (p.332). 

(b) We can calculate the density as follows : 

Take a parallelopiped of cross section unity and length dx with its edges at x and x + dx. 

After the oscillation the edge at x goes to x + ℰ( x) and the edge at x + dx goes to x + d x + ℰ(x + dx)

 Thus the volume of the element (originally dx) becomes

and hence the density becomes 

On substituting we get for the density ρ(x) the curves shown in Fig.(35). referred to above. 

(c) The velocity v (x) at time t = T/4 is

On plotting we get the figure (36).