AS Level Physics 9702: Transverse vs Longitudinal Waves and Intensity ∝ Amplitude²

An earthquake produces two main types of seismic wave that travel through the Earth: P-waves, which are longitudinal, and S-waves, which are transverse.

(a) By referring to the direction of vibration of particles relative to the direction in which a wave travels, distinguish between a transverse wave and a longitudinal wave. [2]

(b) State one everyday example, other than a seismic wave, of (i) a transverse wave, and (ii) a longitudinal wave. [2]

(c) A seismometer close to the epicentre of an earthquake records an S-wave with an amplitude $2.0$ times greater than the amplitude of an S-wave recorded, at the same instant in its oscillation cycle, by a second seismometer much further away, all other factors being equal. Show that the intensity of the wave at the closer seismometer is $4.0$ times the intensity at the more distant seismometer. [2]

(d) At the more distant seismometer, a later S-wave is recorded with an amplitude $3.0$ times smaller than that of the first S-wave measured there. Calculate the ratio of the intensity of this later wave to the intensity of the first wave, both measured at the same (distant) seismometer. [2]

Part (a): Transverse vs longitudinal waves

In a transverse wave, the particles of the medium vibrate perpendicular to the direction in which the wave (and its energy) travels (this is the case for S-waves. In a longitudinal wave, the particles vibrate parallel to (along the same line as) the direction of energy transfer) this is the case for P-waves.

Part (b): Everyday examples

(i) Transverse wave: light (or any other electromagnetic wave), or a wave sent along a stretched string/rope.
(ii) Longitudinal wave: sound travelling through air.

Part (c): Intensity ratio from an amplitude ratio of 2.0

Intensity is proportional to the square of the amplitude: $I \propto A^2$

If the amplitude at the closer seismometer, $A_1$ , is $2.0$ times the amplitude at the more distant seismometer, $A_2$ (i.e. $A_1 = 2.0\,A_2$ ), then: $\frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 = (2.0)^2 = 4.0$

Check (expand directly): $I_1 = kA_1^2 = k(2.0A_2)^2 = k \times 4.0A_2^2 = 4.0\,(kA_2^2) = 4.0\,I_2$ , the same result. This confirms the intensity at the closer seismometer is $\boxed{4.0}$ times the intensity at the more distant one, as required.

Part (d): Intensity ratio from an amplitude ratio of 1/3.0

The later wave’s amplitude, $A_3$ , is $3.0$ times smaller than the first wave’s amplitude at the same (distant) seismometer, $A_2$ , so $A_3 = A_2/3.0$ . Using $I \propto A^2$ again: $\frac{I_3}{I_2} = \left(\frac{A_3}{A_2}\right)^2 = \left(\frac{1}{3.0}\right)^2 = \frac{1}{9}$

Check (as a decimal, computed independently): $\dfrac{1}{3.0} = 0.333...$ , and $0.333...^2 = 0.111... = \dfrac{1}{9}$ , the two methods agree.

So the later wave has $\dfrac{1}{9}$ (about $0.111$ , or roughly $11\%$ ) of the intensity of the first wave at the distant seismometer.

Final answers

(a) Transverse: particle vibration perpendicular to energy transfer; longitudinal: particle vibration parallel to energy transfer
(b) (i) e.g. light; (ii) e.g. sound in air
(c) Intensity ratio $= \boxed{4.0}$
(d) Intensity ratio $= \boxed{1/9} \approx 0.111$