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### Sound Waves

Here are some additional points which may be useful besides the standard discussion of Doppler's Effect.
1. If listener is moving towards the source (irrespective of source moving towards, away or remaining stationary) then the listener is able to get the waves more quickly. In this case the speed of sound for the listener is more. For example in a write-up based question in JEE 2007*, the engine of train A was blowing whistle and the speed of sound for the passenger of the same train would be 340+20=360 m/s.
2. If listener is moving away from the source (irrespective of the source moving towards, away or remaining stationary) then the listener is able to get the waves with some delay (it is as if the listener is running away from them). In this case the speed of sound for the listener is less. For example in a write-up based question in JEE 2007*, the engine of train A was blowing whistle and the speed of sound for the passenger of train B moving in front would be 340-30=310 m/s.
3. When source is moving towards or away from the listener (irrespective of the motion of the listener) then the wavelength changes. If source is not moving wavelength does not change. This is because if the source is moving (say) towards the listener, then the distance between the two consecutive wavefronts would be smaller (second wavefront leaves the source when it already have moved forward). If the source is moving away from the listener, then the distance between the two consecutive wavefronts would be more (second wavefront leaves the source when the source has moved backward).
*Two trains A and B are moving with speeds 20 m/s and 30 m/s respectively in the same direction on the same straight track with B ahead of A. The engines are at the front ends. The engine of train A blows a long whistle. The speed of sound in still air is 340 m/s.

### Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$