14.3. DISPLACEMENT RELATION IN A PROGRESSIVE WAVE

For mathematical description of a travelling wave, we need a function of both position x and time t. Such a function at every instant should give the shape of the wave at that instant. Also, at every given location, it should describe the motion of the constituent of the medium at that location. If we wish to describe a sinusoidal travelling wave the corresponding function must also be sinusoidal.

To mathematically describe a travelling wave, we need to create a formula that changes over time and position. This means for any specific moment in time (t) and at any specific location (x), the formula should represent the wave's shape and the movements of the particles in that medium. If the wave is sinusoidal, implying a smooth, repetitive oscillation, the mathematical representation must also take the sinusoidal form.

Think of a sinusoidal wave as a smooth rippling effect when you drop a stone into a calm pond. At different points in the pond, the water level rises and falls in a sinusoidal pattern, creating ripples which can be mathematically described with sinusoidal functions.

A sinusoidal travelling wave is then described by: ( , ) sin( )= −ω +φy xt a kx t (14.2)

The equation for a sinusoidal travelling wave is given as y(x,t) = a sin(kx - ωt + φ). Here, 'y' represents the displacement of the wave at any position and time, 'a' is the amplitude (the maximum displacement), 'k' is the wave number (related to wavelength), 'ω' is the angular frequency (related to how fast the wave oscillates), and 'φ' is the phase constant (determining the wave's position at time t=0). Each of these components plays a crucial role in shaping the wave's behavior.

Imagine you're pulling a long rope up and down to create waves. The pull (amplitude) determines the height (maximum displacement). How quickly you pull it back and forth (angular frequency) changes how many waves you can create in a given time. The position of your hand when the rope is at rest (phase) determines where the wave starts its journey.

To understand why represents a sinusoidal travelling wave, take a fixed instant, say t = t0. Then, the argument of the sine function is simply kx + constant.

When we analyze the equation further, if we set a specific time (let's say t = t0), the equation simplifies to show how the wave's shape varies based on position x. This means that whether we hold time constant or fix our location, we are able to see how the wave oscillates and moves forward. This relationship gives us valuable insights into wave characteristics, such as how it displaces the medium at different points.

If you've ever recorded a wave's pattern at a water surface, you can relate the height of the wave (displacement) to how far you are from where the stone was dropped (position) at that specific time you took the recording. You would note that as time progresses and you keep 'drawing' the surface, the wave pattern reveals both the ripples (shape) and the locations further from the drop point.

since the sine function varies between 1 and –1, the displacement y (x,t) varies between a and – a. We can take a to be a positive constant, without any loss of generality. Then, a represents the maximum displacement of the constituents of the medium from their equilibrium position. Note that the displacement y may be positive or negative, but a is positive. It is called the amplitude of the wave.

The amplitude (a) indicates the peak value of displacement from the wave's central position (equilibrium). Since the sine function's output ranges from -1 to 1, the wave's displacement can move between its maximum positive (a) and maximum negative (-a) values. This shows how far particles of the medium move up and down as the wave travels.

Consider a child on a swing. The highest point they reach above ground level represents the amplitude. Whether they are swinging up (positive displacement) or down (negative displacement), the highest points are consistent, showing how far they swing out from the center.

The quantity ( kx – ωt + φ) appearing as the argument of the sine function is called the phase of the wave.

The phase (kx - ωt + φ) determines the specific position of the wave at any point in time. By understanding the phase relationship, we can know when and where the wave has maximum displacement, and it helps us track wave motion over time. It essentially captures the timing of the wave's cycle.

Think of the phase like the current time showing when a song plays. Depending on where you are at in the song (the phase), the melody changes. If you come in at different parts, you're experiencing different notes just as different phases of a wave represent different positions in its oscillation.

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