Question bank — Mechanical waves — transverse and longitudinal

False. Each particle only oscillates about its rest spot; it is the pattern (and its energy/momentum) that travels, not the medium's matter.

False. The disturbance is handed on by inter-particle forces (elasticity) and mass (inertia); with no particles there is nothing to pass it along.

False. Air has no shear rigidity, so it cannot support sideways-vibration waves in its bulk — sound in air is purely longitudinal.

True. A solid resists both shear (→ transverse) and compression (→ longitudinal), so both wave types propagate through it.

False. In a non-dispersive medium $v$ is fixed by the medium; raising $f$ just ==shortens $\lambda$== to keep the product constant.

False. That curve plots pressure (or displacement) versus position, not sideways motion — sound stays longitudinal.

False. At a crest the particle is momentarily at rest (turning point of SHM); its speed is maximum at the equilibrium line, where $v_p=A\omega$.

False. Since $\lambda =v/f$ and $v$ is common, the higher-frequency wave has the shorter wavelength.

False. $v\propto \sqrt{T}$, so double tension gives only a factor $\sqrt{2}\approx 1.41$ increase — the square root tames it. (See Wave equation.)

False. Particle velocity is transverse (up/down) for a transverse wave while wave velocity is along $x$; they are perpendicular, and $v_p$ even reverses sign each half cycle.

Error: sound isn't flying particles. Air molecules only jostle their neighbours back and forth; the compression pattern travels, the molecules stay local. (See Sound waves.)

Error: the cork mostly traces a small closed loop and returns; only energy crosses the pool. Any tiny net drift is a second-order effect, not the wave itself.

Error: loudness is amplitude, not frequency or speed; $v$ depends only on the medium and $\lambda$ adjusts to $f$, never to amplitude.

Error: $A\omega$ is the maximum particle speed, an up/down quantity. The wave (pattern) speed is $v=\omega /k=f\lambda$ — a different formula and meaning.

Error: the $+kx$ sign means the phase is constant when $x$ decreases as $t$ increases, so this wave travels in the $-x$ direction. The $+x$-moving wave is $\sin (\omega t-kx)$.

Error: the minus sign is a delay: the particle at $x$ does now what the source did a time $x/v$ ago, so it lags, not leads.

Error: density here is inertia, in the denominator: $v=\sqrt{T/\mu }$, so larger mass-per-length $\mu$ makes the wave slower, not faster.

To pull a neighbour sideways you need a force that opposes shape change (shear); to pull it along the line you only need resistance to being squeezed (compression), which every gas/liquid/solid has.

The medium fixes $v$, and $v=f\lambda$ must hold, so if $f$ rises then $\lambda =v/f$ must fall to keep $v$ constant.

A stronger restoring force snaps particles back faster (numerator) while more inertia makes them sluggish (denominator); the square-root form is what falls out of Newton's law $F=ma$ on a medium element. (See Wave equation.)

Each particle simply copies its neighbour's motion a tiny moment later, so the shape advances even though no particle leaves its own neighbourhood — the stadium "Mexican wave" idea.

Both derivatives of $y=A\sin (\omega t-kx)$ share the same cosine, and their ratio is $\omega /k=v$; the minus sign says a particle on a rising-to-the-right part of the curve is moving downward as the wave slides forward.

$\lambda =v/f$; the same $f$ gives the same $\lambda$ only when $v$ is the same, and $v$ changes with the medium's stiffness and density.

It cannot propagate through the bulk — a fluid has no shear rigidity, so only longitudinal (sound) waves survive in the interior. (Surface water waves are a special mixed case, not bulk transverse.)

No meaningful disturbance remains: $y\to 0$ everywhere, so there is no energy to transport even though $\omega$, $k$, $v$ are still defined.

$v=\sqrt{T/\mu }\to 0$ — with no restoring force the disturbance cannot be handed on, so no wave travels.

Yes — each travelling wave keeps its own $v=f\lambda$; their sum forms a standing pattern, but that superposition doesn't change either wave's speed. (See Standing waves & resonance and Superposition and Interference.)

Speed is maximum ($A\omega$) and acceleration is zero, because restoring force $\propto -y$ vanishes at $y=0$ — the SHM signature. (See Simple Harmonic Motion.)

No — $v$ is fixed by the air; the observed frequency shifts (Doppler) while the wave speed through the medium stays the same. (See Doppler effect.)

The frequency is set by the source and is continuous, but $v$ drops (larger $\mu$) so $\lambda =v/f$ shortens in the heavier string.