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d. A wave moving in one direction reflects at a barrier and interferes with the incoming wave moving in the opposite direction. If the displacements of the first and second waves are given as:
(i) Find the resulting wave [tex]\((y_1 + y_2)\)[/tex]. (1.5 marks)
(ii) If in (i) above [tex]\( y_1 = 4 \cos (2t - 30) \)[/tex] and [tex]\( y_2 = -3 \sin (2t + 60) \)[/tex], then find the new resulting wave. (1.5 marks)
We need to find the resulting wave [tex]\(y_{\text{result}} = y_1 + y_2\)[/tex].
So the resulting wave is simply the sum of the two given displacements: [tex]\[
y_{\text{result}} = 4 \cos (2t - 30) + (-3 \sin (2t + 60))
\][/tex] [tex]\[
y_{\text{result}} = 4 \cos (2t - 30) - 3 \sin (2t + 60)
\][/tex]
In conclusion, the resulting waves are: 1. For part (i): [tex]\[
y_{\text{result}} = 50 \sin (15t) \cos (6x)
\][/tex]
2. For part (ii): [tex]\[
y_{\text{result}} = 4 \cos (2t - 30) - 3 \sin (2t + 60)
\][/tex]
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