At one instant a right-travelling sound wave has u=A sin(2πx/λ). Identify displacement and whether compression/rarefaction is greatest at x=0, λ/4, λ/2 and 3λ/4. Explain using displacements of neighbouring particles why a maximum in the displacement graph is not a maximum compression. State what a displacement–time graph of one particle measures. Two additional displacement-distance graphs are supplied: T is a transverse wave and L is a longitudinal wave. Read the amplitude and wavelength of each from the graphs. Explain what a positive vertical-axis value means for a particle in each wave, relative to the wave's direction of travel. Identify the x-position of greatest compression in L and explain why the positive crest in T is not a compression. Accessible graph data: both horizontal axes show x/m from 0 to 4, ticks 0,1,2,3,4; each vertical scale runs −2 to +2 mm, ticks at each integer. Graph T is y/mm and graph L is u/mm. Each smooth sinusoidal curve passes through (0,0), (1,+2), (2,0), (3,−2), (4,0), with propagation to the right (+x).

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At x0: u0, rarefaction; λ/4: +A, neither extreme density; λ/2: u0, compression; 3λ/4: −A, neither extreme density. Compression depends on particles converging, not on one particle's displacement. The time graph gives amplitude and period, not wavelength. Both amplitudes are 2.0 mm and both wavelengths 4.0 m. Positive T displacement is perpendicular to travel; positive L displacement is along +x. L has maximum compression at x = 2.0 m, where relative displacements bring neighbours closer together; T's positive crest is a transverse displacement, not longitudinal compression. Read displacements 0,+A,0,−A. At x=0 a positive spatial displacement gradient means neighbours are farther apart than at equilibrium, giving rarefaction; this is not a statement about their instantaneous velocities. At x=λ/2 a negative spatial displacement gradient means neighbours are closer together than at equilibrium, giving compression. At displacement extrema local slope is zero, so pressure/density disturbance is zero in the small-amplitude model. Explain the fixed-position time axis and distinguish T from λ. Read amplitude 2.0 mm for both graphs (1). Read wavelength 4.0 m for both graphs (1). Contrast perpendicular displacement in T with parallel displacement in L (1). Identify L compression at 2.0 m using relative displacements and distinguish the transverse crest from compression (1).

Interpret a longitudinal displacement graph

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