Question

A sinusoidal wave moving along a string is shown twice in the figure, as crest $A$ travels in the positive direction of an $x$ axis by distance $d=6.0\, cm$ in $4.0\, m$. The tick marks along the axis are separated by $10\, cm$; height $H=6.00\, mm$. The wave equation is

• A. $y=(3 mm ) \sin \left[16 x-2.4 \times 10^{2} t\right]$
• B. $y=(3 mm ) \sin \left[16 x+2.4 \times 10^{2} t\right]$
• C. $y=(3 mm ) \sin \left[8 x+2.4 \times 10^{2} t\right]$
• D. $y=(3 mm ) \sin \left[8 x-2.4 \times 10^{2} t\right]$

Answer 1

Answer: (A)

The amplitude $A$ is half of the $6.00\, mm$ vertical range shown in the figure, that is, $A=3.0\, mm$.
The speed of the wave is $v=d / t=15\, m / s$,
where $d=0.060\, m$ and $t=0.0040\, s$.
The angular wave number is $k=2 \pi / \lambda$ where $\lambda=0.40\, m$.
Thus, $k=\frac{2 \pi}{\lambda}=16\, rad / m$.
The angular frequency is found from
$\omega=k v=(16\, rad / m )(15\, m / s )=2.4 \times 10^{2} rad / s$
We choose the minus sign (between $k x$ and $\omega t$ ) in the argument of the sine function because the wave is shown traveling to the right (in the $+x$ direction). Therefore, with SI units understood, we obtain
$y=A \sin [k x-\omega t]=(3\, mm ) \sin \left[16 x-2.4 \times 10^{2} t\right]$