Do Sinusoidal Equations Represent Different Wave Types?

[Amith2006]

Sir,
1)Is it true that y = a sin(wt) is the equation of a wave pulse and y = a sin(wt – kx) is the equation of a progressive wave?
I think it is true.
2)Will the following 2 waves produce stationary waves?
y1 = a cos(kx – wt)
y2 = a sin(kx + wt)
I think the answer is No. What do you say Sir?

 

[maverick280857]

Sir,
1)Is it true that y = a sin(wt) is the equation of a wave pulse and y = a sin(wt – kx) is the equation of a progressive wave?

$y(0,t)=a\sin(\omega t)$ is the wave as seen from $x=0$. If you were to measure the vertical displacement $y(0,t)$ then you would find it to vary simple harmonically.

A wave exists for all x and t whereas a pulse is a portion of a wave which exists for some time, much like the wave pulse generated when you shake one end of a string (the other end being fixed) before it strikes the end and standing waves are set up. To describe a pulse, you need to describe the wavefunction as $y(x,t)$ for some region when it is "on" and zero elsewhere. So you need to include the span of the wave pulse.

2)Will the following 2 waves produce stationary waves?
y1 = a cos(kx – wt)
y2 = a sin(kx + wt)
I think the answer is No. What do you say Sir?

Why don't you add the waves and see for yourself?? Its not that difficult and the best way to figure out is to add them yourself rather than have someone tell you the answer.

 

[kyle8921]

Yes, it is true that y = a sin(wt) is the equation of a wave pulse and y = a sin(wt – kx) is the equation of a progressive wave. In the first equation, the wave is only traveling in one direction, while in the second equation, the wave is traveling in both directions.

As for the second question, the two waves y1 and y2 will not produce stationary waves as they have different frequencies and directions of propagation. Stationary waves are formed when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. In this case, the two waves have different frequencies and are traveling in different directions, so they will not produce stationary waves.