| A transverse wave propagating along x-axis is represented by: [AIPMT (S) 2006] |
| \[y\,(x,t)=8.0\,\sin \,\left( 0.5\,\pi x-4\pi t-\frac{\pi }{4} \right)\] |
| where x is in metres and t is in seconds. The speed of the wave is: |
A) \[4\,\pi \,m/s\]
B) \[0.5\,\pi \,m/s\]
C) \[\frac{\pi }{4}\,m/s\]
D) \[8\,\,m/s\]
Correct Answer: D
Solution :
| Key Idea: The standard transverse wave propagating along x-axis can be written as |
| \[y=a\sin (kx-\omega t+\phi )\] |
| The given equation is |
| \[y(x,\,t)=8.0\,\sin \,\left( 0.5\,\pi \,x-4\pi t-\frac{\pi }{4} \right)\] (i) |
| The standard wave equation can be written as, |
| \[y=a\sin (kx-\omega t+\phi )\] ...(iii) |
| where a is amplitude, k the propagation constant and \[\omega \] the angular frequency, comparing the Eqs. (i) and (ii), we have |
| \[k=0.5\,\pi ,\,\omega =4\pi \] |
| \[\therefore \] Speed of transverse wave |
| \[v=\frac{\omega }{k}=\frac{4\,\pi }{0.5\,\pi }\] |
| \[=8\text{ }m/s\] |
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