Question

A continuous sinusoidal longitudinal wave is sent along a very long coiled spring from an attached oscillating source. The wave travels in the negative direction of an x axis; the source frequency is \[25Hz\]; at any instant the distance between successive points of maximum expansion in the spring is \[24cm\]; the maximum longitudinal displacement of a spring particle is \[0.30cm\]; and the particle at \[x = 0\] has zero displacement at time \[t = 0\]. If the wave is written in the form \[s(x,t) = {s_m}\cos (kx \pm \omega t)\], what is the correct choice of sign in front of \[\omega \]?

Answer 1

Hint: In this question, we know the direction of wave propagation in a medium and the equation of wave. So, we can compare this with the standard equation of wave to understand the direction which will determine the sign.

Complete step by step answer:
The equation of wave is given as \[s(x,t) = {s_m}\cos (kx \pm \omega t)\] and in the question it is mentioned that the wave is travelling in negative x-axis direction , so the equation of wave will become
\[s(x,t) = {s_m}\cos (kx + \omega t)\]
Hence the correct sign in front of \[\omega \]will be positive.

Additional information:
The waves can be classified into two types on the basis of direction of medium particles relative to wave propagation direction. They are longitudinal and transverse waves.
Longitudinal waves have a distinct feature that is the direction of the particles of medium is the same as the direction or we can say parallel to the direction of motion of this wave. They produce compression (part where the particles are close) and rarefaction (where particles are spread). Example is sound waves.
In transverse waves, the movement of particles of medium is in perpendicular direction to the wave propagation. Example is electromagnetic waves or waves in ropes. They form crests (maximum displacement in positive direction) and trough (maximum displacement in negative direction). They are produced in solids generally. They cannot be generated in gases.
The distance between two successive crests or two successive troughs in case of transverse wave and in longitudinal, the distance between two successive compression or two successive rarefactions is called wavelength of that particular wave.

Note:
A longitudinal wave y at position x and time t can be represented in the form of wave equation as
$y(x,t) = {y_0}\cos (\omega t \pm kx)$
where y is the displacement of the wave, x is the distance travelled by the wave from the source and $\omega $ is the angular velocity of the wave.