Question

Draw a picture of two waves, one with three times the wavelength of the other. Label the wave with the longer wavelength.

Short answer

The picture should show two waves. The first wave should be of some arbitrary wavelength (\( \lambda \)), and the second wave should have three times the wavelength of the first one (3\( \lambda \)). The second wave should be labeled as the wave with the longer wavelength.

Step-by-step solution

Draw the first wave

Draw a wave across the page. A method to do this is to start by drawing a straight horizontal line, and then drawing a series of semi-circular arcs along the line, alternating them above and below the line, to create a wave pattern. Ensure the arcs are of the same size to represent equal wavelengths.

Identify the wavelength of the first wave

Wavelength is the distance between one wave cycle. Measure the length from one peak directly to the next peak (or one trough to the next corresponding trough). This will be the wavelength of the first wave. It’s also helpful to illustrate this on the diagram by drawing a double-headed arrow over this distance and labelling it \( \lambda \).

Draw the second wave

Now, draw a second wave beneath the first one with three times the wavelength of the first wave. To do this, make the distance between the peaks (or troughs) on the second wave three times the length of the wavelength of the first wave. Again, it’s beneficial to indicate this on the diagram by drawing a double-headed arrow over this distance and labelling it '3\( \lambda \)'.

Label the wave with the longer wavelength

Finally, label the second wave as the wave with the longer wavelength. This can be done by writing 'Longer Wavelength' beside the second wave.

Key concepts

Wave Pattern

Understanding the wave pattern is crucial for students who are exploring the basics of wave behavior. A wave pattern depicts the oscillating movement of energy through a medium, such as water or air. It's like watching the ripples created when a stone is thrown into a pond. Each ripple represents a wave, and if you look closely, you can see a repetitive rise and fall, known as the crests (peaks) and troughs (valleys).

To visualize this concept, imagine a line of children holding hands and moving together to form a human wave at a sports event. As each child stands up and sits down, they create a moving pattern. In the exercise, this pattern is represented by semi-circular arcs that alternate above and below a baseline, resembling a smooth, undulating landscape with hills and valleys.

Measuring Wavelength

Measuring wavelength is akin to measuring the length of a single ripple from one peak to another in our pond analogy. In practice, wavelength, usually denoted by the Greek letter lambda ( )), is the distance over which the wave's shape repeats. It is one of the fundamental properties of waves and is measured in meters, centimeters, or other units of length.

In the provided exercise, you visualize this by drawing one wave cycle and marking the distance between consecutive crests or troughs. To measure this accurately, imagine stretching a string along this path or using a ruler in an actual diagram, ensuring the string or ruler is parallel to the baseline of the wave pattern. The length of this segment represents the wave's wavelength, and by using the same measurement increment, you can draw a second wave with a wavelength three times that of the first, giving a clear representation of their relative sizes.

Wave Properties

Wave properties encompass more than just wavelength. They include frequency, amplitude, speed, and energy. Frequency refers to the number of wave cycles that pass a certain point per second. Amplitude is the height from the rest position (the baseline of your wave pattern) to the crest, which indicates how much energy the wave is carrying; higher waves mean higher energy.

Wave speed determines how fast the wave propagates through the medium and is calculated by multiplying the wavelength ( 0)) by the frequency (f). Moreover, these properties interrelate. For example, when the wavelength of a wave is increased while keeping the speed constant, the frequency decreases. This is conceptually represented in the exercise where the wave with a longer wavelength, labelled 'Longer Wavelength, has fewer cycles over the same distance compared to the wave with a shorter wavelength, allowing students to visualize how changing one property affects the others.