For the wave described in Exercise 14.8, plot the displacement ($y$) versus ($t$) graphs for $x = 0$, $2$ and $4\ \text{cm}$. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase?

For the wave described in Exercise 14.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4cm. Wha

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Concept understanding — Wave Type Classification

Wave Type Classification: From Intuition to Precision

Imagine you're standing at the edge of a still pond. You drop a pebble. Ripples spread outward in circles. Now imagine you're holding one end of a long rope tied to a wall. You flick your wrist once — a single hump travels down the rope, hits the wall, and comes back. These are both waves, but they behave differently. Why?

The key difference lies in what is moving versus what is waving.

The Core Intuition

A wave is a disturbance that carries energy from one place to another without permanently moving the medium itself. But the direction of that disturbance relative to the wave's travel direction gives us our first major classification.

Think of a crowd at a stadium doing "the wave." People stand up and sit down (the disturbance moves up-down), but the wave itself travels around the stadium (left-right). The motion of each person is perpendicular to the wave's travel. That's one type.

Now think of a slinky stretched on a table. If you push one end toward the other, a compression travels along the slinky. Each coil moves along the same line as the wave — forward and back. That's the other type.

The Precise Classification

Waves are classified into two fundamental types based on the relationship between the direction of particle displacement (how the medium moves) and the direction of wave propagation (which way the energy travels).

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Transverse Waves

In a transverse wave, the particles of the medium oscillate perpendicular to the direction the wave travels.

Example: Light waves (electromagnetic waves), waves on a string, water waves (surface component), seismic S-waves
Key feature: The wave has crests (high points) and troughs (low points)
Visual: Think of a rope shaken up and down — the rope moves vertically, the wave moves horizontally

Longitudinal Waves

In a longitudinal wave, the particles of the medium oscillate parallel to the direction the wave travels.

Example: Sound waves in air, seismic P-waves, slinky compressions
Key feature: The wave has compressions (regions of high density) and rarefactions (regions of low density)
Visual: Think of a slinky pushed and pulled — coils bunch up and spread out along the same line the wave moves

Note

Some waves are neither purely transverse nor purely longitudinal. Water waves, for instance, have particles moving in circular paths — a combination of both. These are called surface waves or Rayleigh waves in seismology.

Why This Matters

This classification isn't just academic. It determines:

What materials a wave can travel through: Transverse waves (like light) can travel through vacuum. Longitudinal waves (like sound) need a medium. But mechanical transverse waves (like rope waves) also need a medium — the distinction is about how the medium moves, not whether a medium exists. …

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This question reuses the travelling wave from Exercise 14.8, $y(x,t)=3.0\sin(36t+0.018x+\pi/4)$ (amplitude in cm, $t$ in s, $x$ in cm). At each fixed $x=0,2,4\ \text{cm}$ , the $y$ - $t$ graph is a sine curve of the same amplitude ( $3.0\ \text{cm}$ ) and same period ( $T=2\pi/36\approx0.175\ \text{s}$ ), shifted only slightly by a tiny extra phase -- the oscillatory motion differs from point to point only in phase, never in amplitude or frequency.

Setting up: the graph at a fixed position

For a travelling wave $y(x,t)=A\sin(\omega t+kx+\phi_0)$ , fixing $x$ turns the two-variable function into a pure function of time alone. Everything about the graph's shape -- amplitude $A$ and angular frequency $\omega$ -- is unchanged by which $x$ you pick; only the constant phase offset $kx+\phi_0$ shifts with position.

Applying it to this wave

The wave from Exercise 14.8 is $y(x,t) = 3.0\sin(36t + 0.018x + \pi/4)$ , so $A=3.0\ \text{cm}$ , $\omega=36\ \text{rad/s}$ , $k=0.018\ \text{rad/cm}$ , base phase $\phi_0=\pi/4$ .

Period (same at every point):

T = \frac{2\pi}{\omega} = \frac{2\pi}{36} \approx 0.175\ \text{s}

Phase offset at each given $x$ :

$x=0$ : $\phi = \pi/4 \approx 0.785\ \text{rad}$
$x=2\ \text{cm}$ : $\phi = 0.036+\pi/4 \approx 0.821\ \text{rad}$
$x=4\ \text{cm}$ : $\phi = 0.072+\pi/4 \approx 0.857\ \text{rad}$

Since $k=0.018\ \text{rad/cm}$ is very small (wavelength $\lambda=2\pi/k\approx349\ \text{cm}$ , much larger than these few centimetres), the phase changes only slightly from $x=0$ to $x=4\ \text{cm}$ .

Shapes of the graphs …

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Setting up: the graph at a fixed position

Applying it to this wave

Shapes of the graphs …

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