Question

here is the question: If z = f(s+at) for some function f(x) find and simplify (d^2z)/(dt^2) - (a^2)(d^2z)/(ds^2) = ? or here is the link to the question https://docs.google.com/open?id=0BwO0tVQUoc6yQWRMU19vOWN5NE0

Answer 1

Let z = f(u) where u = s + at
\[\frac{dz}{du}=f'(u)\]
\[\frac{\delta u}{\delta s}=1\]
\[\frac{\delta u}{\delta t}=a\]
\[\frac{\delta z}{\delta s}=\frac{dz}{du}\times \frac{\delta u}{\delta s}\]
\[\frac{\delta z}{\delta s}=f'(u)\times 1=f'(u)\]
\[\frac{\delta z}{\delta t}=\frac{dz}{du}\times \frac{\delta u}{\delta t}=f'(u)\times a\]
\[\frac{\delta ^{2}z}{\delta t ^{2}}=f''(u)\times a\]
\[\frac{\delta ^{2}z}{\delta s ^{2}}=f''(u)\]
\[\frac{\delta ^{2}z}{\delta t ^{2}}-a ^{2}\frac{\delta ^{2}z}{\delta s ^{2}}=(f''(u)\times a)-a ^{2}f''(u)=\]
f''(s + at) * a(1-a)