Intro to Waves: Videos & Practice Problems

Waves are disturbances that travel through a medium, such as sound through air or ripples in water. There are two main types: transverse waves, where particle displacement is perpendicular to wave motion, and longitudinal waves, where displacement is parallel. Key concepts include wavelength, defined as the distance between successive crests or compressions, and amplitude, the height of the wave. The relationship between wave speed (v), wavelength (λ), and frequency (f) is given by the equation v = λf.

Intro to Waves and Wave Speed

Waves are defined as disturbances that travel through a medium, which is a material or substance that facilitates the wave's movement. Common examples include the ripples created when a rock is dropped in water, the vibrations of a string, or sound waves traveling through air. Understanding the types of waves is crucial, as they can be categorized into two main types: transverse and longitudinal waves.

In a transverse wave, the displacement of particles in the medium occurs perpendicular to the direction of wave motion. For instance, if you move a slinky up and down, the wave travels horizontally while the particles move vertically. Conversely, in a longitudinal wave, the particle displacement is parallel to the wave motion. This can be visualized by pushing and pulling a slinky back and forth, where the wave moves in the same direction as the particle movement.

Key characteristics of waves include wavelength, amplitude, and period. The wavelength (\(\lambda\)) is the distance between successive crests or troughs in a transverse wave, or between compressions in a longitudinal wave. The amplitude is the maximum height of the wave from its rest position, which is half the vertical distance of the wave. The period (T) is the time it takes for one complete cycle of the wave to pass a given point, and frequency (f) is the number of cycles per second, related to the period by the equation \(f = \frac{1}{T}\).

For both types of waves, the speed (v) can be calculated using the relationship between wavelength and frequency, expressed as:

\[ v = \lambda f \]

This equation indicates that the speed of a wave is equal to its wavelength multiplied by its frequency. For example, if sound travels at a speed of 343 meters per second and has a frequency of 260 Hz, the wavelength can be calculated by rearranging the equation to:

\[ \lambda = \frac{v}{f} = \frac{343 \, \text{m/s}}{260 \, \text{Hz}} \approx 1.32 \, \text{m} \]

Understanding these fundamental concepts of waves, including their types, characteristics, and the relationships between speed, wavelength, and frequency, is essential for further studies in physics and related fields.

A wave has a wavelength of 10mm and moves at a speed of 50mm/s. What is the frequency of this wave?

• A. f = 50 Hz/s
• B. f = 5 Hz
• C. f = 500 s

Answer: (B)

Example 1

In this problem, we analyze a wave on a horizontal rope, focusing on the vertical and horizontal distances between the crest and trough. The vertical distance from the crest (the highest point of the wave) to the trough (the lowest point) is given as 15 centimeters. This distance is crucial for determining the amplitude of the wave.

The amplitude, defined as the height of the wave from the midline (or equilibrium position) to the crest, is half of the total vertical distance from crest to trough. Therefore, the amplitude can be calculated as:

Amplitude = \(\frac{1}{2} \times \text{(Crest to Trough Distance)} = \frac{1}{2} \times 15 \, \text{cm} = 7.5 \, \text{cm}\)

To convert this to SI units, we can express it as:

Amplitude = \(0.075 \, \text{m}\)

Next, we need to determine the wavelength, denoted by the Greek letter lambda (\(\lambda\)). The wavelength is defined as the distance between two consecutive identical points on the wave, such as crest to crest or trough to trough. The problem states that the horizontal distance from crest to trough is 20 centimeters. However, this distance only represents half of the wavelength, as a complete wave cycle includes both the descent to the trough and the ascent back to the next crest.

Thus, the full wavelength can be calculated as:

\(\lambda = 2 \times \text{(Crest to Trough Distance)} = 2 \times 20 \, \text{cm} = 40 \, \text{cm}\)

In SI units, this is expressed as:

\(\lambda = 0.4 \, \text{m}\)

In summary, for the wave described, the amplitude is \(0.075 \, \text{m}\) and the wavelength is \(0.4 \, \text{m}\). Understanding these concepts is essential for analyzing wave behavior in various physical contexts.

Here’s what students ask on this topic:

What is the difference between transverse and longitudinal waves?

Transverse and longitudinal waves differ primarily in the direction of particle displacement relative to wave motion. In transverse waves, particle displacement is perpendicular to the direction of wave propagation. An example is a wave on a string, where the string moves up and down while the wave travels horizontally. In contrast, longitudinal waves have particle displacement parallel to the wave direction. Sound waves in air are a common example, where air particles compress and rarefy in the same direction the wave travels. Understanding these differences is crucial for analyzing wave behavior in various mediums.

How do you calculate wave speed?

Wave speed (v) can be calculated using the equation:

$v=\lambda f$

where $\lambda$ is the wavelength and $f$ is the frequency. Wavelength is the distance between successive crests or compressions, and frequency is the number of wave cycles per second. For example, if a sound wave has a frequency of 260 Hz and travels at 343 m/s, the wavelength can be found by rearranging the equation:

$\lambda =\frac{v}{f}$

Substituting the values:

$\lambda =\frac{343}{260}=1.32 m$

What is the relationship between wavelength and frequency?

The relationship between wavelength ($\lambda$) and frequency ($f$) is inversely proportional, as described by the wave speed equation:

$v=\lambda f$

Here, $v$ is the wave speed. If the wave speed is constant, an increase in frequency results in a decrease in wavelength, and vice versa. For example, if a wave travels at 300 m/s and has a frequency of 150 Hz, the wavelength is:

$\lambda =\frac{300}{150}=2 m$

This inverse relationship is fundamental in wave mechanics.

What is the amplitude of a wave and how is it measured?

The amplitude of a wave is the maximum displacement of particles from their equilibrium position. It is a measure of the wave's energy and is typically measured from the wave's midpoint (equilibrium position) to its peak (crest) or trough. For transverse waves, amplitude is the vertical distance from the equilibrium to the crest or trough. For example, if a wave oscillates between 0 and 2 meters, the amplitude is 2 meters. In longitudinal waves, amplitude is less commonly discussed but can be considered as the maximum displacement of particles in the direction of wave propagation.

What is the period of a wave and how is it related to frequency?

The period (T) of a wave is the time it takes for one complete cycle of the wave to pass a given point. It is measured in seconds. The period is inversely related to the frequency (f), which is the number of cycles per second. This relationship is given by:

$f=\frac{1}{T}$

or equivalently:

$T=\frac{1}{f}$

For example, if a wave has a frequency of 5 Hz, its period is:

$T=\frac{1}{5}=0.2 s$

Understanding this relationship helps in analyzing wave properties.