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A string of length \( L \) fixed at both ends vibrates in its fundamental mode at a frequency \( v \) and a maximum
\( \mathrm{P} \) amplitude \( A \). (a) Find the wavelength and the wave number \( k \). (b) Take the origin at one end of the string
W and the \( X \)-axis along the string. Take the \( Y \)-axis along the direction of the displacement. Take \( t=0 \) at the instant when the middle point of the string passes through its mean position and is going towards the positive \( y \)-direction. Write the equation describing the standing wave.
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