A sinusoidal wave of wavelength 7.5 \ mathrmcm$7.5 \ \mathrm{cm}$ travels a distance of 1.2 \ mathrmcm$1.2 \ \mathrm{cm}$ along the x-direction in 0.3 \ mathrmsec$0.3 \ \mathrm{sec}$. The crest P is at x = 0$x = 0$ at t = 0 \ mathrmsec$t = 0 \ \mathrm{sec}$ and maximum displacement of the wave is 2 \ mathrmcm$2 \ \mathrm{cm}$ . Which equation correctly represents this wave?

### Related Formula 1. Wave function (moving along +x$+x$ direction) with a peak at x=0, t=0$x=0, t=0$: y(x, t) = A cos(kx - omega t)$$y(x, t) = A \cos(kx - \omega t)$$ 2. Wave number: k = frac2pilambda$$k = \frac{2\pi}{\lambda}$$ 3. Wave speed: v = fracomegak$$v = \frac{\omega}{k}$$ ### Core Logic Given parameters: - Wavelength lambda = 7.5 \ mathrmcm$\lambda = 7.5 \ \mathrm{cm}$ - Distance travelled Delta x = 1.2 \ mathrmcm$\Delta x = 1.2 \ \mathrm{cm}$ in Delta t = 0.3 \ mathrms$\Delta t = 0.3 \ \mathrm{s}$ - Maximum displacement (amplitude) A = 2 \ mathrmcm$A = 2 \ \mathrm{cm}$ Let's calculate the wave parameters: 1. **Wave number (k$k$):** k = frac2pi7.5 = frac20pi75 = frac4pi15 approx 0.838 \ mathrmrad/cm$$k = \frac{2\pi}{7.5} = \frac{20\pi}{75} = \frac{4\pi}{15} \approx 0.838 \ \mathrm{rad/cm}$$ 2. **Wave speed (v$v$):** v = fracDelta xDelta t = frac1.20.3 = 4 \ mathrmcm/s$$v = \frac{\Delta x}{\Delta t} = \frac{1.2}{0.3} = 4 \ \mathrm{cm/s}$$ 3. **Angular frequency (omega$\omega$):** omega = v cdot k = 4 times frac4pi15 = frac16pi15 approx 3.35 \ mathrmrad/s$$\omega = v \cdot k = 4 \times \frac{4\pi}{15} = \frac{16\pi}{15} \approx 3.35 \ \mathrm{rad/s}$$ Since the crest is at x=0$x=0$ at t=0$t=0$, y(0,0) = 2 = A$y(0,0) = 2 = A$. This boundary condition demands a cosine function. ### Step 1: Write wave equation Substitute A$A$, k$k$, and omega$\omega$ into the standard form: y(x, t) = 2 cos(0.83x - 3.35t) \ mathrmcm$$y(x, t) = 2 \cos(0.83x - 3.35t) \ \mathrm{cm}$$ This perfectly matches Option (1). ### Pattern Recognition Sees: Wavelength and speed to determine travelling wave equation. Trap: Choosing sine instead of cosine. Since the crest (maximum displacement) is at x=0, t=0$x=0, t=0$, y(0,0)$y(0,0)$ must equal A$A$, which is satisfied only by the cosine function. Shortcut: Calculate k = 2pi / 7.5 approx 0.83$k = 2\pi / 7.5 \approx 0.83$. This immediately eliminates Options (3) and (4). Calculate speed v = 4$v = 4$, so omega = 4 times 0.83 approx 3.35$\omega = 4 \times 0.83 \approx 3.35$, which points directly to Option (1). ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Waves