Question

just question 3. question 1 only helps for answer in question 3

A string is stretched between two posts with the equilibrium position of the string lying along the x-axis. After the string is plucked, let z = s(x, t) denote the vertical displacement of the string at position and time t. string in equilibrium plucked string 1. Give a short interpretation, in words, of each of the following quantities: i) The function (2,0). ii) The partial derivative iii) The partial derivative iv) The second partial derivative v) The second partial derivative

Answer 1

Answer:

a) The wave equation is a second order differential equation (partial differential equation) of both space and time. Thus, solving it we need two boundary conditions (they are known as Cauchy and Newmann conditions).

b) Suppose, $s(x,t)=f(x-ct)$

$\therefore \frac{\partial ^{2}s}{\partial x^{2}}=\frac{\partial ^{2}f}{\partial x^{2}}$ & $\frac{\partial ^{2}s}{\partial t^{2}}=c^{2 }\frac{\partial ^{2}f}{\partial t^{2}}$

which implies is a solution of wave equation.

c) Suppose, say $s(x,t)=e^{x^{2}+ct^{2}}$

Here also $\frac{\partial ^{2}s}{\partial x^{2}}$ &  $\frac{\partial ^{2}s}{\partial t^{2}}$ exists but this doesn't solve wave equation.

d) Suppose, $s(x,t)=Acos(kct)cos(kx+t)$

This solves the wave equation but this is not travelling wave. This is known as standing wave. The amplitude depends upon time. At some point x that satisfy $cos(kx+t)=0$

there is no displacement at any time, such points are called nodes.

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