Show that the equation of motion of identical atoms of mass M in a chain, M dt2 d²Un = K (Un+1 + Un-1 – 2Un) , ecomes the continuum elastic wave equation, with vs as the sound velocity for the long wavelengths, a²U dx?

Calc based physics The general wave function on a string is given by : Y(x,t) = Acos(Kx+wt+Q). Prove that the ratio of d2Y/dt2 over d2Y/dx2 is (w/k)2. Prove that w/k represents velocity.

Snapshot: This a snapshot at t = 2s of the wave function: Calculate or read from the graph: y(x,t) = Axsin 2 × sin [2x (1-1)] a) Amplitude A, the wavelength A and the wave number k. b) Determine all possible periods T consis- tent with the graph. For the largest pe- riod T calculate the speed of the propaga- tion v. In which direction does the wave propagate (in +x or -x)? c) What is the maximal speed and accelera- tion of the points along the wave? y(m) 2 1 1 2 3 4 x(m)

A string of 1 m length clamped at both ends is plucked in the middle to generate a standing wave. Take the frequency of the first harmonic to be 10 Hz and the amplitude of the oscillations to be 2 cm. Consider the motion to be a simple harmonic one where appropriate.  Calculate the value of maximum velocity of up and down motion of the string at midpoint. Provide your anwser in m/s.