jmorgan said:Hi, I can't quite understand how to do this question please could someone help :)
Show, by the principle of superposition, that
u(x, t) =
∞
∑ An sin(npix)e2n2pi2t
n=1
where A1, A2,..., are arbitrary constants.
Thanks
The equation for showing superposition of a wave is u(x, t) = A sin(kx - wt), where A is the amplitude of the wave, k is the wave number, x is the position, t is the time, and w is the angular frequency.
Superposition refers to the combination of two or more waves to create a new wave, while interference refers to the interaction of two or more waves resulting in a pattern of constructive and destructive interference.
The phase of a wave refers to the position of the wave at a specific point in time. In superposition, the phase of each individual wave determines how they combine to form the resulting wave.
In constructive interference, the waves combine to create a larger amplitude, while in destructive interference, the waves cancel each other out, resulting in a smaller or zero amplitude.
Superposition can be observed in many real-life phenomena, such as the interference patterns in water waves, sound waves, and light waves. It can also be observed in musical instruments, where multiple sound waves combine to create a single tone.