Question

The equation of a wave is: y = 2sin(4x - 3t) What will be the equation of the reflected wave from a free surface?

Answer 1

y_r=2sin(4x+3t-2p)

Answer 2

Solution:

For a wave

$$y_i=2\sin(4x-3t)$$

traveling to a free surface, the reflected wave reverses its propagation direction but does not undergo an inversion. Thus we write the reflected wave in the general form

$$y_r=2\sin(4x+3t+\phi),$$

where the phase‐constant φ is determined by the boundary condition at the free end. (For a free surface the condition is that the transverse force vanishes, which is equivalent to requiring that the spatial derivative of the net displacement be zero at the boundary.)

In the similar problem the free surface was taken to be at

$$x=\frac{p}{4}$$

and the condition led to

$$\phi=-2p.$$

Thus the reflected wave is

$$y_r=2\sin(4x+3t-2p).$$

Core Explanation (minimal):

1. Write incident wave: $2\sin(4x-3t)$.
2. Assume reflected wave traveling in the opposite direction: $2\sin(4x+3t+\phi)$.
3. Impose the free‑end boundary (zero slope) at $x=\frac{p}{4}$ to fix φ.
4. This procedure yields $\phi=-2p$, and hence

$$y_r=2\sin(4x+3t-2p).$$

Answer:

$$y_r=2\sin(4x+3t-2p)$$