Transverse And Longitudinal Wave Practice Answer Key

Transverse and Longitudinal Waves: Practice Problems and Solutions

Understanding transverse and longitudinal waves is crucial for grasping fundamental concepts in physics. This comprehensive guide provides a detailed explanation of both wave types, followed by a series of practice problems with complete, step-by-step solutions. Whether you're a student preparing for an exam or simply seeking a deeper understanding of wave phenomena, this resource will equip you with the knowledge and tools to master this important topic.

What are Transverse Waves?

Transverse waves are characterized by the oscillation of particles perpendicular to the direction of wave propagation. Imagine a wave traveling along a rope; as the wave moves horizontally, the rope itself moves up and down, perpendicular to the wave's direction. Key features of transverse waves include:

• Crest: The highest point of the wave.
• Trough: The lowest point of the wave.
• Amplitude: The maximum displacement of a particle from its equilibrium position.
• Wavelength (λ): The distance between two consecutive crests or troughs.
• Frequency (f): The number of complete oscillations per unit time (usually measured in Hertz, Hz).
• Period (T): The time taken for one complete oscillation. The relationship between frequency and period is: T = 1/f.
• Wave speed (v): The speed at which the wave propagates. The relationship between wave speed, frequency, and wavelength is: v = fλ.

Examples of Transverse Waves:

• Light waves: Electromagnetic waves, including visible light, are transverse waves.
• Waves on a string: Plucking a guitar string generates transverse waves.
• Seismic S-waves: These secondary waves generated during earthquakes are transverse waves.

What are Longitudinal Waves?

In contrast to transverse waves, longitudinal waves involve the oscillation of particles parallel to the direction of wave propagation. Think of a sound wave traveling through air; the air molecules vibrate back and forth in the same direction as the wave's movement. Key features of longitudinal waves include:

• Compression: Regions of high particle density.
• Rarefaction: Regions of low particle density.
• Amplitude: The maximum displacement of a particle from its equilibrium position. This corresponds to the difference in pressure between a compression and a rarefaction.
• Wavelength (λ): The distance between two consecutive compressions or rarefactions.
• Frequency (f): The number of complete oscillations per unit time (Hz).
• Period (T): The time taken for one complete oscillation (T = 1/f).
• Wave speed (v): The speed at which the wave propagates (v = fλ).

Examples of Longitudinal Waves:

• Sound waves: Sound waves travel through various media (air, water, solids) as longitudinal waves.
• Seismic P-waves: These primary waves generated during earthquakes are longitudinal waves.
• Ultrasound waves: Used in medical imaging and other applications.

Practice Problems: Transverse Waves

Problem 1: A transverse wave on a string has a frequency of 10 Hz and a wavelength of 0.5 meters. What is the speed of the wave?

Solution:

We use the formula v = fλ.

• f = 10 Hz
• λ = 0.5 m

Therefore, v = (10 Hz)(0.5 m) = 5 m/s. The speed of the wave is 5 m/s.

Problem 2: A transverse wave travels along a rope at a speed of 20 m/s. If the wavelength is 2 meters, what is the frequency of the wave?

Solution:

We rearrange the formula v = fλ to solve for frequency: f = v/λ.

• v = 20 m/s
• λ = 2 m

Therefore, f = (20 m/s) / (2 m) = 10 Hz. The frequency of the wave is 10 Hz.

Problem 3: A transverse wave has an amplitude of 0.1 meters and a wavelength of 1 meter. Sketch the wave, labeling the crest, trough, amplitude, and wavelength.

Solution:

(This problem requires a graphical representation. You would draw a sine wave with a peak (crest) at 0.1 meters above the equilibrium line and a trough at 0.1 meters below. The distance between two consecutive crests or troughs would be labeled as 1 meter (wavelength), and the distance from the equilibrium line to the crest (or trough) would be labeled as 0.1 meters (amplitude))

Practice Problems: Longitudinal Waves

Problem 4: A longitudinal sound wave has a frequency of 440 Hz and a wavelength of 0.77 meters. What is the speed of the sound wave?

Solution:

We use the formula v = fλ.

• f = 440 Hz
• λ = 0.77 m

Therefore, v = (440 Hz)(0.77 m) ≈ 338.8 m/s. The speed of the sound wave is approximately 338.8 m/s.

Problem 5: A longitudinal wave travels through a solid at a speed of 5000 m/s. If the frequency is 2500 Hz, what is the wavelength?

Solution:

We rearrange the formula v = fλ to solve for wavelength: λ = v/f.

• v = 5000 m/s
• f = 2500 Hz

Therefore, λ = (5000 m/s) / (2500 Hz) = 2 m. The wavelength of the wave is 2 meters.

Problem 6: Describe the difference in particle motion between a longitudinal wave and a transverse wave. Give examples of each.

Solution:

In a transverse wave, particles oscillate perpendicular to the direction of wave propagation. Examples include light waves and waves on a string. In a longitudinal wave, particles oscillate parallel to the direction of wave propagation. Examples include sound waves and seismic P-waves.

Comparing Transverse and Longitudinal Waves: A Table

Advanced Concepts and Applications

Understanding the basic principles of transverse and longitudinal waves opens the door to exploring more advanced concepts like:

• Superposition: When two or more waves meet, their displacements add together. This can lead to constructive interference (waves add up to a larger amplitude) or destructive interference (waves cancel each other out).
• Diffraction: The bending of waves around obstacles.
• Refraction: The bending of waves as they pass from one medium to another.
• Doppler Effect: The change in frequency of a wave due to the relative motion between the source and the observer.

These advanced concepts have significant applications in various fields, including:

• Medical Imaging: Ultrasound uses longitudinal waves to create images of internal organs.
• Communication: Radio waves (transverse waves) are used for wireless communication.
• Seismology: Seismic waves (both transverse and longitudinal) are used to study the Earth's interior.
• Acoustics: The study of sound waves (longitudinal) and their interaction with the environment.

This comprehensive guide provides a solid foundation for understanding transverse and longitudinal waves. By working through the practice problems and exploring the advanced concepts, you can develop a strong grasp of this essential physics topic. Remember to practice regularly and seek additional resources if needed. Mastering wave mechanics will significantly enhance your understanding of many important physical phenomena.