Question
Given below are some functions of x and t to represent the displacement of an elastic wave.$\text{y}=100\cos(100\pi\text{t+0.5x})$
✓
Answer
$\text{y}=4\sin\Big(5\text{x}-\frac{\text{t}}{2}\Big)+3\cos\Big(5\text{x}-\frac{\text{t}}{2}\Big)$$\text{Let}4=\text{a}\cos\phi\ ...(\text{ii})$and $3=\text{a}\sin\phi\ ...(\text{iii})$
$\text{a}^2\cos^2\phi+\text{a}^2\sin^2\phi=4^2+3^2$ Squaring and adding (ii), (iii)
$\text{a}^2=25\text{K}\Rightarrow\text{a}=5$
Substituting (ii), (iii) in (i)
$\text{y}=\text{a}\cos\phi\sin\Big(5\text{x}-\frac{\text{t}}{2}\Big)+\text{a}\sin\phi\cos\Big(5\text{x}-\frac{\text{t}}{2}\Big)$
$\text{y}=\text{a}\sin\Big(5\text{x}-\frac{\text{t}}{2}+\phi\Big)$
$\text{y}=\text{5}\sin\Big(5\text{x}-\frac{\text{t}}{2}+\phi\Big)$
Which represents the progressive wave in $+\text{x}$ direction as the sign of Kx (or5x) and $\omega\text{t}\Big(\frac{1}{2}\text{t}\Big)$ are opposite so it travels in $+\text{x}$ direction. So (d) (ii)
$\text{a}^2\cos^2\phi+\text{a}^2\sin^2\phi=4^2+3^2$ Squaring and adding (ii), (iii)
$\text{a}^2=25\text{K}\Rightarrow\text{a}=5$
Substituting (ii), (iii) in (i)
$\text{y}=\text{a}\cos\phi\sin\Big(5\text{x}-\frac{\text{t}}{2}\Big)+\text{a}\sin\phi\cos\Big(5\text{x}-\frac{\text{t}}{2}\Big)$
$\text{y}=\text{a}\sin\Big(5\text{x}-\frac{\text{t}}{2}+\phi\Big)$
$\text{y}=\text{5}\sin\Big(5\text{x}-\frac{\text{t}}{2}+\phi\Big)$
Which represents the progressive wave in $+\text{x}$ direction as the sign of Kx (or5x) and $\omega\text{t}\Big(\frac{1}{2}\text{t}\Big)$ are opposite so it travels in $+\text{x}$ direction. So (d) (ii)
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