Progressive waves

Introduction to Progressive Waves

Welcome to the study of waves! If you’ve ever seen ripples spreading across a pond or felt the vibration of a loud bass speaker, you have already experienced progressive waves. In this chapter, we are going to explore how waves travel, how we measure them, and the clever ways they carry energy from one place to another. Don't worry if it seems like a lot of new vocabulary at first—we will break it down piece by piece!

1. What is a Progressive Wave?

A progressive wave is a disturbance that travels through a medium (like air, water, or a rope) or a vacuum, transferring energy from one point to another without transferring any matter.

Think of it like this: Imagine a "stadium wave" at a sports match. The people stand up and sit down (the vibration), but they don't move to the other side of the stadium. Only the pattern moves across the crowd. That's exactly how waves work!

How we see waves in action:

• Ropes: If you flick one end of a rope, a "hump" travels to the other end. The rope stays in your hand, but the energy reaches the wall.
• Springs (Slinkys): You can push and pull a spring to see a pulse move through it.
• Ripple Tanks: These use water to show how waves spread out across a surface.

Key Takeaway: Progressive waves move energy, not matter.

2. The Language of Waves (Key Terms)

To master this chapter, you need to be comfortable with these terms. Think of these as the "coordinates" for describing any wave.

• Displacement (\(x\)): The distance of a point on the wave from its equilibrium (rest) position. It can be positive or negative.
• Amplitude (\(A\)): The maximum displacement. It’s the height of a "crest" or the depth of a "trough" measured from the middle.
• Wavelength (\(\lambda\)): The distance between two identical points on consecutive waves (e.g., crest to crest). Measured in meters (m).
• Period (\(T\)): The time taken for one complete wave to pass a point. Measured in seconds (s).
• Frequency (\(f\)): The number of complete waves passing a point per unit time. Measured in Hertz (Hz). Relationship: \(f = \frac{1}{T}\).
• Wave Speed (\(v\)): The distance the wave travels per unit time.

What about Phase Difference?

Phase difference tells us how "out of sync" two points on a wave are. We measure it in degrees (\(^\circ\)) or radians.
- If two points are doing the exact same thing at the same time, they are in phase (phase difference = \(0^\circ\)).
- If one is at a crest while the other is at a trough, they are completely out of phase (phase difference = \(180^\circ\) or \(\pi\) radians).

Quick Review Box:
High Frequency = More waves per second = Shorter Period.
Large Amplitude = More Energy.

3. The Wave Equation

There is one very important formula that links speed, frequency, and wavelength. Let's see where it comes from!

1. Speed is \(\text{Distance} / \text{Time}\).
2. In the time of one period (\(T\)), the wave travels a distance of one wavelength (\(\lambda\)).
3. So, \(v = \frac{\lambda}{T}\).
4. Since \(f = \frac{1}{T}\), we can swap them to get the Wave Equation:

\(v = f\lambda\)

Example: If a wave has a frequency of \(100\text{ Hz}\) and a wavelength of \(2\text{ m}\), its speed is \(100 \times 2 = 200\text{ m s}^{-1}\).

Key Takeaway: If the speed of a wave is constant (like light in a vacuum), then increasing the frequency must decrease the wavelength.

4. Using a Cathode-Ray Oscilloscope (CRO)

A CRO is basically a fancy voltmeter that draws a graph of Voltage against Time. It is very useful for "seeing" sound waves.

How to find Amplitude and Frequency:

1. Amplitude: Look at the "Y-gain" (or Y-sensitivity) setting (e.g., \(2\text{ V/cm}\)). Measure the height of the wave on the screen in cm and multiply by the Y-gain.

2. Frequency:
- Look at the "Time-base" setting (e.g., \(5\text{ ms/cm}\)).
- Measure the horizontal distance for one full wave (the period \(T\)) in cm.
- Multiply the distance by the time-base to get the time in seconds.
- Use \(f = \frac{1}{T}\) to find the frequency.

Common Mistake: Watch your units! Time-bases are often in milliseconds (ms) or microseconds (\(\mu\)s). Convert them to seconds before calculating frequency!

Did you know? The CRO doesn't show the actual wave moving through the air; it shows the electrical signal created by a microphone catching that wave.

5. Energy and Intensity

As a wave travels, it carries energy. The rate at which this energy arrives is related to Intensity.

Definition: Intensity (\(I\)) is the power passing through a unit area perpendicular to the direction of wave travel.
Formula: \(I = \frac{P}{A}\) (where \(P\) is Power and \(A\) is Area).

The "Square Law" Relationship

This is a favorite topic for exam questions! The intensity of a wave is directly proportional to the square of its amplitude.

\(I \propto A^2\)

What this means:
- If you double the amplitude (\(\times 2\)), the intensity increases by \(2^2\), which is 4 times.
- If you triple the amplitude (\(\times 3\)), the intensity increases by \(3^2\), which is 9 times.

Key Takeaway: Even a small increase in amplitude leads to a much larger increase in intensity (brightness for light, or loudness for sound).

6. Summary of Progressive Waves

Don't worry if this seems tricky at first; practice with the formulas will make it feel like second nature! Here is what you must remember:

• Waves transfer energy without transferring matter.
• Use \(v = f\lambda\) for all wave calculations.
• Frequency is the reciprocal of the period (\(f = \frac{1}{T}\)).
• Intensity is proportional to the square of the amplitude (\(I \propto A^2\)).
• Use the time-base on a CRO to find the period and then the frequency.