Two sinusoidal waves of identical amplitude and wavelength propagate to the right…

PHYSICS · Multi-part · A Level · Physics · Waves · Waves

Problem

Two sinusoidal waves of identical amplitude A=2.0 cmA = 2.0\,\text{cm} and wavelength λ=0.40 m\lambda = 0.40\,\text{m} propagate to the right along the same stretched string with identical speed. At time t=0t=0 their individual displacements are described by the functions y1(x,0)=A sin⁡ ⁣(2πλx),y2(x,0)=A sin⁡ ⁣(2πλx+2π3),y_1(x,0)=A\,\sin\!\left(\tfrac{2\pi}{\lambda}x\right),\qquad y_2(x,0)=A\,\sin\!\left(\tfrac{2\pi}{\lambda}x+\tfrac{2\pi}{3}\right), where xx is measured from the left end of the string. Answer the following questions about the net displacement within the single interval 0≤x≤λ0\le x\le\lambda at t=0t=0. (a) Determine every position in the interval 0≤x≤λ0\le x\le\lambda where the instantaneous net displacement of the string is a local maximum in magnitude. Express each position as a fraction of λ\lambda. Identify whether the displacement there is positive or negative. (b) Determine every position in the same interval where the instantaneous net displacement is exactly zero. Express each position as a fraction of λ\lambda. (c) Explain, without performing further calculations, why the positions you found in parts (a) and (b) respectively correspond to constructive and destructive interference for these two waves. (d) Sketch the resulting waveform for 0≤x≤λ0\le x\le\lambda at t=0t=0. Clearly label the positions of the largest positive displacement, the largest negative displacement, and all points of zero displacement.

Related theory

Browse the problem library