After two seconds pulses will overlap each other.
NOTE : According to superposition principle the string will not have any distortion and will be straight.
Hence there will be no P.E. The total energy will be only kinetic.

$$\eqalign{ & {\text{According to question,}} \cr & {y_1} = a\sin \left( {\omega t + kx + 0.57} \right) \cr & {\text{and}}\,\,{y_2} = a\cos \left( {\omega t + kx} \right) \cr & {\text{or}}\,\,{y_2} = a\sin \left( {\frac{\pi }{2} + \omega t + kx} \right) \cr & {\text{As phase difference,}} \cr & \Delta \phi = {\phi _2} - {\phi _1} = \frac{\pi }{2} - 0.57 \cr & = 1.57 - 0.57 = 1\,rad \cr} $$

Frequency of Sound heard by the man from approaching train.
$${n_a} = n\left( {\frac{v}{{v - {v_s}}}} \right) = 240\left( {\frac{{320}}{{320 - 4}}} \right) = 243\,Hz$$
Frequency of sound heard by the man from receding train $${n_r} = n\left( {\frac{v}{{v + {v_s}}}} \right) = 240\left( {\frac{{320}}{{320 + 4}}} \right) = 237\,Hz$$
Hence, number of beats heard by man per $$\sec$$
$$ = {n_a} - {n_r} = 243 - 237 = 6$$

Interference phenomenon is common to sound and light. In sound, the interference is said to be constructive at points where resultant intensity is maximum (are in phase) and destructive at points where resultant intensity is minimum or zero (are in opposite phase).

Frequency of wave is a function of the source of waves. Therefore, it remains unchanged.

Two wave are said to be coherent, when they have same frequency, amplitude and constant phase difference.

$$\eqalign{ & {\text{Given}}\,\,\frac{{nv}}{{2\ell }} = 315\,\,{\text{and}}\,\,\left( {n + 1} \right)\frac{v}{{2\ell }} = 420 \cr & \Rightarrow \frac{{n + 1}}{n} = \frac{{420}}{{315}} \Rightarrow n = 3 \cr & {\text{Hence}}\,\,3 \times \frac{v}{{2\ell }} = 315 \Rightarrow \frac{v}{{2\ell }} = 105\,Hz \cr} $$
The lowest resonant frequency is when $$n = 1$$
Therefore lowest resonant frequency $$= 105\,Hz.$$