Four sound waves are to be sent through the same tube of air, in the same direction:

$\begin{array}{l}{\mathbf{s}}_{\mathbf{1}}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{t}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\mathbf{9}\mathbf{.00}\mathbf{\text{\hspace{0.17em}nm}}\mathbf{)}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{2}\mathbf{\pi }\mathbf{x}\mathbf{}\mathbf{-}\mathbf{700}\mathbf{\pi }\mathbf{t}\mathbf{)}\\ {\mathbf{s}}_{\mathbf{2}}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{t}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\mathbf{9}\mathbf{.00}\mathbf{\text{\hspace{0.17em}nm}}\mathbf{)}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{2}\mathbf{\pi }\mathbf{x}\mathbf{}\mathbf{-}\mathbf{}\mathbf{700}\mathbf{\pi }\mathbf{t}\mathbf{}\mathbf{+}\mathbf{0}\mathbf{.7}\mathbf{\pi }\mathbf{)}\\ {\mathbf{s}}_{\mathbf{3}}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{t}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\mathbf{9}\mathbf{.00}\mathbf{\text{\hspace{0.17em}nm}}\mathbf{)}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{2}\mathbf{\pi }\mathbf{x}\mathbf{}\mathbf{-}\mathbf{700}\mathbf{\pi }\mathbf{t}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\pi }\mathbf{)}\\ {\mathbf{s}}_{\mathbf{4}}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{t}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\mathbf{9}\mathbf{.00}\mathbf{\text{\hspace{0.17em}nm}}\mathbf{)}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{2}\mathbf{\pi }\mathbf{x}\mathbf{}\mathbf{-}\mathbf{700}\mathbf{\pi }\mathbf{t}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{.7}\mathbf{\pi }\mathbf{)}\mathbf{.}\end{array}$

What is the amplitude of the resultant wave? (Hint:Use a phasor diagram to simplify the problem.)

Equations of four sound waves,

i) $s_1(x,t)=\left(9.00\text{\hspace{0.17em}nm}\right)\cos (2\pi x-700\pi t)$

ii) $s_2(x,t)=\left(9.00\text{\hspace{0.17em}nm}\right)\cos (2\pi x-700\pi t+0.7\pi )$

iii) $s_3(x,t)=\left(9.00\text{\hspace{0.17em}nm}\right)\cos (2\pi x-700\pi t+\pi )$

iv) $s_4(x,t)=\left(9.00\text{\hspace{0.17em}nm}\right)\cos (2\pi x-700\pi t+1.7\pi )$

Using the superposition of waves, we can find the amplitude of the resultant wave. The concept of finding the resultant wave using the phase difference cancellation method is useful.

According to the principle of superposition of waves, if two waves have the phase difference of $\pi$radians, then there is destructive interference, and the amplitude of the resultant wave will be zero.

If we see the above given waves, we note that

1^st and 3^rd wave differ bysimilarly, 2^nd and 3^rd wave differ by.

So the resultant amplitude of the above waves is zero.

If we give names to those waves $S_1,S_2,S_3,S_4$as A, B, C, D respectively, then the phase diagram of those waves will be as below: