Transverse and longitudinal waves

Transverse waves. Particle motion is perpendicular to the direction the wave travels. A pulse sent along a stretched string is the classic example: each bit of string moves up and down while the pulse itself moves horizontally. Crests (highest points) and troughs (lowest points) are visible, and the distance between two neighbouring crests is one wavelength.

Longitudinal waves. Particle motion is parallel to the direction of travel — particles shuffle back and forth along the same line the wave is moving on. Sound in air is the standard example: air molecules do not "ride" the sound to your ear; they oscillate locally, alternately bunching together into compressions (higher pressure and density) and spreading apart into rarefactions (lower pressure and density). One wavelength is the distance from one compression to the next compression.

The four quantities every wave needs. Amplitude $A$ is the maximum displacement of a particle from its rest position. Time period $T$ is the time for one particle to complete one full oscillation. Frequency $f=1/T$ counts oscillations per second. Wavelength $\lambda$ is the distance between two particles in identical phase — crest to crest, or compression to compression.

Wave speed. In one period $T$ the whole wave pattern advances exactly one wavelength $\lambda$, so speed is simply distance over time:

where: $v$ is wave speed (m/s), $\lambda$ is wavelength (m), $T$ is time period (s), and $f=1/T$ is frequency (Hz). This relation holds for both transverse and longitudinal waves alike — it comes from the definition of wavelength and period, not from what is oscillating.

The travelling-wave equation. A source oscillating as $y=A\sin(\omega t)$ sends out a wave whose displacement depends on both position $x$ and time $t$:

where: $A$ is amplitude, $k=\dfrac{2\pi}{\lambda}$ is the propagation constant (rad/m), $\omega=\dfrac{2\pi}{T}=2\pi f$ is angular frequency (rad/s), and the minus sign marks travel in the $+x$ direction (a plus sign, $y=A\sin(kx+\omega t)$, marks travel in the $-x$ direction). General form: $y=A\sin(kx\pm\omega t)$. Because $v=\omega/k=\dfrac{2\pi f}{2\pi/\lambda}=f\lambda$, this equation is fully consistent with the speed relation above — it is just the same physics written with angles instead of distances.

A sound wave (longitudinal) in air has wavelength $1.65\ \text{m}$ and frequency $200\ \text{Hz}$ — the same frequency as the transverse string wave used earlier in this topic. Find (a) its speed, (b) its angular frequency $\omega$, (c) its propagation constant $k$.

Step 1. Speed from $v=f\lambda$: $v = 200 \times 1.65 = 330\ \text{m/s}$, the familiar speed of sound in air — notice this is over three times the $100\ \text{m/s}$ found for the transverse string wave at the same frequency in the Pass-1 worked example, because the two waves travel in different media with different wavelengths.

Step 2. Angular frequency depends only on frequency, not on wave type: $\omega = 2\pi f = 2\pi \times 200 = 1256.6\ \text{rad/s}$ — identical to the string wave's $\omega$, since both waves share $f=200\ \text{Hz}$.

Step 3. Propagation constant: $k=\dfrac{2\pi}{\lambda}=\dfrac{2\pi}{1.65}=3.807\ \text{rad/m}$.

Step 4. Check: $v=\omega/k=1256.6/3.807=330.1\ \text{m/s}$ ✓, matching part (a) to rounding. The sound wave travels at 330 m/s, with $\omega=1256.6\ \text{rad/s}$ and $k=3.807\ \text{rad/m}$.

• Treating $v=f\lambda$ as a "speed of the particle" — it is the speed of the wave pattern; individual particles only oscillate about a fixed point and their own instantaneous velocity ($\partial y/\partial t$) is a completely different, constantly changing quantity.
• Assuming wavelength cannot be defined for a longitudinal wave because "there is no up-down curve to measure" — it is still the distance between two consecutive identical-phase points, just measured between compressions (or rarefactions) instead of crests.
• Mixing up the sign convention in $y=A\sin(kx\pm\omega t)$: the minus sign means travel in $+x$, the plus sign means travel in $-x$ (for this way of writing $kx$ first) — guessing from intuition is unreliable, so check the sign directly.
• Forgetting that $\omega$ and $f$ depend only on the source's oscillation rate, while $k$ and $\lambda$ additionally depend on the medium through $v$ — two waves of the same frequency in different media have the same $\omega$ but different $k$ and $\lambda$, exactly as in the worked example above.

Ask the tutor: Want to practice extracting $A$, $k$, $\omega$, $\lambda$, $f$ and direction from a few more wave equations, or need the sign-convention rule drilled until it is automatic?