Question

While deriving the formula of fundamental frequency for a string fixed at both ends, the two waves coming from both ends are taken as Asin(kx−wt)

and Asin(kx+wt+ϕ).
After applying the boundary condition that y=0
at x=0
, we get that ϕ=0
. but according to my book, standing waves are obtained when waves are produced at one fixed end and then they interfere with the reflected wave. But the reflected wave is inverted, right? Doesn't this mean that the phase difference between the interferering waves if π
? How can it be 0?

If I take the interfering waves as Asin(wt−kx)
and Asin(wt+kx+ϕ)
instead, it shouldn't make any differece, right? I've just cahnged the intitial phase. But now, after applying the boundary condition, I get ϕ=π
.

The incoming wave is taken as Asin(kx-wt) and the reflected wave is taken as Asin(kx+wt+ϕ
). But how can we use the same 'x' in both the equations? The second equation gives the displacement at a distance x from the end and the x in first equations is measured from the origin. I'm guessing that the equatuon of the displacement produced by reflected wave should be Asin(k(L-x)+wt+ϕ
)

There's one more but it's related to sound waves:

The formula for the apparent wavelength of sound waves when the source is moving is λ′=v−uvλ
which is constant as v
, u
and λ
are constants. Bu there is a diagram in my book showing the spherical wavefronts originating from a moving point source. In the diagram, the distance between successisive wavefronts decreases with time and hence wavelength varies.

Answer 1

In the case of standing waves on a string fixed at both ends, the reflected wave is indeed inverted compared to the incident wave. The phase difference between the interfering waves depends on the boundary condition at the fixed end of the string.

Step-by-step explanation:

In the case of standing waves on a string fixed at both ends, the reflected wave is indeed inverted compared to the incident wave. However, the phase difference between the interfering waves is not necessarily always pi (π). The phase difference depends on the boundary condition at the fixed end of the string.

When the wave encounters a fixed boundary, such as in the case of a string fixed at both ends, the reflected wave is 180° (π radians) out of phase with respect to the incident wave. This means that the phase difference between the interfering waves is π, not 0.

The formula for the fundamental frequency of a standing wave on a string fixed at both ends is given by f = (1/2L) * sqrt(T/μ), where L is the length of the string, T is the tension in the string, and μ is the linear mass density of the string.