question_answer 61)

                  Given be low are some functions of \[x\] and \[t\] to represent to represent the displacement of an elastic wave.                 (a) \[y=5\,\cos \,(4x)\,\sin \,(20\,t)\]                 (b) \[y=5\,\sin \,(5x=t/2)+3\cos \,(5x-t/2)\]                 (c) \[y=10\,\cos \,[(252-250\,\pi ]\]\[\cos [(252+250)\pi t]\]                 (d) \[y=100\,\cos \,(100\pi t+0.5\,x)\]                 State which of these represent                 (a) a travelling wave along \[-x\] direction                 (b) a stationary wave                  (d) a travelling wave along \[+x\] direction.                 Given reasons for your answers.

Answer:

                  (i) A travelling wave along +x direction is given by                 \[y=a\sin \,(kx-\omega t)\]                 or \[y=a\,\cos (kx-\omega t)\]                 (ii) A travelling wave along \[-x\] direction                 \[y=a\sin \,(kx+\omega t)\]                 or \[y=a\cos \,(kx+\omega t)\]                 (iii) A stationary wave is represented by                 \[y=(2a\sin \,kx)\cos \omega t\]                 or \[y=(2a\cos \,kx)\sin \omega t\]                 (iv) Beats are represented by                 \[y=2a\,\cos \,2\pi \left( \frac{{{v}_{1}}-{{v}_{2}}}{2} \right)t\,\sin \,2\pi \left( \frac{{{v}_{1}}+{{v}_{2}}}{2} \right)\]                 Now compare the given equations with these standard equations. (a) \[y=5\,\cos \,(4x)\,\sin \,(20t)\] represents a stationary wave. (b) \[y=4\,\sin \,\left( 5x-\frac{t}{2} \right)\,+3\,\cos \,\left( 5x\,-\frac{t}{2} \right)\]                 is the superposition of two travelling waves along \[+x\] direction.                 (c) \[y=\,10\,\cos \,[(252-250)\pi t]\cos \,[(252+250)\pi t]\]                 represents beats                 (d) \[y=\,100\,\cos \,(100\pi t+0.5x)\] represents a travelling wave along \[-x\] direction.