Athin taut string is fixed at both ends and stretched along the horizontal x-axis with its left end at x = 0. it is vibrating in its third overtone, and the equation for the vertical displacement of any point on the string is y(x, t) = (1.22 cm) sin[(14.4 m-1)x] cos[(166 rad/s)t]. (a) what are the frequency and wavelength of the fundamental mode of this string? (b) how long is the string? (c) how fast do waves travel on this string?

(a) Wavelength is 0.436 m

(b) Length is 0.872 m

(c) 11.518 m/s

Solution:

As per the question:

The eqn of the displacement is given by:

         (1)

n = 4

Now,

We know the standard eqn is given by:

          (2)

Now, on comparing eqn (1) and (2):

A = 1.22 cm

K =

where

A = Amplitude

K = Propagation constant

= angular velocity

Now, to calculate the string's wavelength,

(a)

where

K = propagation vector

(b) The length of the string is given by:

(c)  Now, we first find the frequency of the wave:

Now,

Speed of the wave is given by: