A wave travelling along a string is described by,
$$y(x, t) = 0.005 \sin(80.0\,x - 3.0\,t),$$
in which the numerical constants are in SI units ($0.005\ ext{m}$, $80.0\ ext{rad m}^{-1}$, and $3.0\ ext{rad s}^{-1}$). Calculate (a) the amplitude, (b) the wavelength, and (c) th

Question

A wave travelling along a string is described by,
$$y(x, t) = 0.005 \sin(80.0\,x - 3.0\,t),$$
in which the numerical constants are in SI units ($0.005\ 	ext{m}$, $80.0\ 	ext{rad m}^{-1}$, and $3.0\ 	ext{rad s}^{-1}$). Calculate (a) the amplitude, (b) the wavelength, and (c) the period and frequency of the wave. Also, calculate the displacement $y$ of the wave at a distance $x = 30.0\ 	ext{cm}$ and time $t = 20\ 	ext{s}$?

Accepted Answer

> [!TLDR]
> Reading the wave $y=0.005\sin(80.0x-3.0t)$ directly against the standard form $A\sin(kx-\omega t)$ gives amplitude $A=0.005\ 	ext{m}$, wavelength $\lambda\approx0.0785\ 	ext{m}$, period $T\approx2.094\ 	ext{s}$, frequency $f\approx0.477\ 	ext{Hz}$, and a displacement of $y\approx\boxed{+4.96	imes10^{-3}\ 	ext{m}}$ at $x=30.0\ 	ext{cm}$, $t=20\ 	ext{s}$.

## Reading the wave's parameters

The wave $y(x,t)=0.005\sin(80.0x-3.0t)$ is in the standard form $y=A\sin(kx-\omega t)$:
- Amplitude: $A=0.005\ 	ext{m}$ (5 mm)
- Wave number: $k=80.0\ 	ext{rad/m}$
- Angular frequency: $\omega=3.0\ 	ext{rad/s}$

## (a) Amplitude

$$A = 0.005\ 	ext{m}$$

## (b) Wavelength

$$\lambda = \frac{2\pi}{k} = \frac{2\pi}{80.0} \approx 0.0785\ 	ext{m}\ (7.85\ 	ext{cm})$$

## (c) Period and frequency

$$T = \frac{2\pi}{\omega} = \frac{2\pi}{3.0} \approx 2.094\ 	ext{s}, \qquad f = \frac1T = \frac{\omega}{2\pi} \approx 0.477\ 	ext{Hz}$$

## Displacement at $x=30.0\ 	ext{cm}$, $t=20\ 	ext{s}$

Convert $x$ to metres: $30.0\ 	ext{cm} = 0.300\ 	ext{m}$.
$$y = 0.005\sin\big(80.0	imes0.300 - 3.0	imes20\big) = 0.005\sin(24.0-60.0) = 0.005\sin(-36.0)$$

Since $\sin(-	heta)=-\sin	heta$:
$$y = -0.005\sin(36.0\ 	ext{rad})$$

Reduce $36.0$ rad modulo $2\pi$ ($2\pi\approx6.2832$):
$$36.0 - 5	imes6.2832 = 4.584\ 	ext{rad}$$
This angle lies between $\pi\ (3.1416)$ and $\dfrac{3\pi}2\ (4.7124)$ -- the third quadrant, where sine is **negative**. Numerically, $\sin(4.584)\approx-0.9918$, so $\sin(36.0)\approx-0.9918$.

Substituting back: …

Q.A wave travelling along a string is described by,
[!FORMULA] $y(x, t) = 0.005 \sin(80.0\,x - 3.0\,t),$
in which the numerical constants are in SI units ( $0.005\ \text{m}$ , $80.0\ \text{rad m}^{-1}$ , and $3.0\ \text{rad s}^{-1}$ ). Calculate

Concept understanding — Wave Type Classification

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Why This Matters

Reading the wave's parameters

(a) Amplitude

(b) Wavelength

(c) Period and frequency

Displacement at $x=30.0\ \text{cm}$ , $t=20\ \text{s}$

Reading the wave's parameters

(a) Amplitude

(b) Wavelength

(c) Period and frequency

Displacement at x=30.0 cmx=30.0\ \text{cm}x=30.0 cm, t=20 st=20\ \text{s}t=20 s

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Displacement at $x=30.0\ \text{cm}$ , $t=20\ \text{s}$

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