Wave On A String Answer Key

Wave on a String: A Comprehensive Guide with Answers

Understanding wave propagation on a string is fundamental to grasping many physics concepts, from musical instruments to telecommunications. This comprehensive guide explores the physics behind wave motion on a string, providing detailed explanations, worked examples, and answers to common questions. We’ll delve into the crucial factors influencing wave speed, frequency, wavelength, and the generation of various wave patterns. By the end, you'll have a solid foundation for tackling more advanced wave phenomena.

What is a Transverse Wave on a String?

A transverse wave on a string is a disturbance that travels along the string, with the particles of the string moving perpendicular to the direction of wave propagation. Imagine plucking a guitar string – the string vibrates up and down (perpendicular motion), while the wave itself travels along the length of the string. This is in contrast to a longitudinal wave, where particle motion is parallel to the wave's direction.

Key Characteristics of Transverse Waves on Strings:

• Amplitude: The maximum displacement of a particle from its equilibrium position. A larger amplitude means a more energetic wave.
• Wavelength (λ): The distance between two consecutive points in the wave that are in the same phase (e.g., two consecutive crests or troughs).
• Frequency (f): The number of complete oscillations (cycles) per unit time, typically measured in Hertz (Hz).
• Period (T): The time it takes for one complete cycle of the wave. It's the reciprocal of frequency (T = 1/f).
• Wave Speed (v): The speed at which the wave travels along the string. This is crucial and depends on the properties of the string itself.

Factors Affecting Wave Speed on a String

The speed of a transverse wave on a string is determined by two primary factors:

• Tension (T): The force applied along the length of the string. Higher tension leads to a faster wave speed.
• Linear Mass Density (μ): The mass per unit length of the string. A lighter string (lower μ) will allow waves to travel faster.

The relationship between these factors and wave speed is given by the following equation:

v = √(T/μ)

This equation is a cornerstone of understanding wave propagation on strings. It reveals that increasing tension increases the wave speed, while increasing the linear mass density decreases it.

Calculating Wave Parameters: Worked Examples

Let's work through some examples to illustrate how to apply the concepts and equations we've discussed.

Example 1: A string with a linear mass density of 0.01 kg/m is under a tension of 100 N. Calculate the speed of a wave traveling on this string.

Solution:

Using the equation v = √(T/μ), we have:

v = √(100 N / 0.01 kg/m) = √(10000 m²/s²) = 100 m/s

Therefore, the wave speed is 100 m/s.

Example 2: A wave on a string has a frequency of 50 Hz and a wavelength of 0.2 m. What is its speed?

Solution:

The speed of a wave is given by the equation:

v = fλ

v = 50 Hz * 0.2 m = 10 m/s

The wave speed is 10 m/s.

Example 3: A string with a linear mass density of 0.005 kg/m has a wave traveling on it at 200 m/s. What is the tension in the string?

Solution:

Rearranging the wave speed equation to solve for tension (T):

T = μv²

T = 0.005 kg/m * (200 m/s)² = 200 N

The tension in the string is 200 N.

Standing Waves on a String: Nodes and Antinodes

When a wave reflects from the end of a string (either fixed or free), it can interfere with the incident wave. If the conditions are right, this interference creates a standing wave, also known as a stationary wave. These standing waves exhibit points of zero displacement called nodes and points of maximum displacement called antinodes.

Harmonics and Fundamental Frequency

Standing waves on a string only exist at specific frequencies called harmonics or overtones. The lowest frequency harmonic is called the fundamental frequency or first harmonic. The fundamental frequency depends on the length (L) of the string, its tension (T), and its linear mass density (μ).

The fundamental frequency (f₁) is given by:

f₁ = (1/2L)√(T/μ)

Higher harmonics are integer multiples of the fundamental frequency:

fₙ = nf₁, where n = 1, 2, 3... (n represents the harmonic number)

Types of Boundary Conditions: Fixed and Free Ends

The boundary conditions at the ends of the string significantly affect the pattern of the standing waves.

• Fixed End: The end of the string is fixed, and the displacement is always zero at that point. This results in a node at the fixed end.

• Free End: The end of the string is free to move, resulting in an antinode at that end. This is often achieved using a very light pulley system.

Solving Problems with Standing Waves: Examples

Let's tackle some problems involving standing waves.

Example 4: A string of length 1 meter, linear mass density 0.002 kg/m, and tension 50 N is fixed at both ends. Find the frequency of the fundamental mode of vibration.

Solution:

Using the fundamental frequency equation:

f₁ = (1/2L)√(T/μ) = (1/(2 * 1 m))√(50 N / 0.002 kg/m) ≈ 50 Hz

The fundamental frequency is approximately 50 Hz.

Example 5: A string fixed at both ends vibrates in its third harmonic (n=3) at a frequency of 150 Hz. If the length of the string is 0.8 m, what is the speed of the wave?

Solution:

The frequency of the nth harmonic is given by: fₙ = nf₁ = n(v/2L).

We know f₃ = 150 Hz and n = 3. We can solve for v:

v = (2Lf₃)/n = (2 * 0.8 m * 150 Hz)/3 = 80 m/s

The speed of the wave is 80 m/s.

Energy and Power in Wave Propagation

Waves on a string carry energy. The energy is proportional to the square of the amplitude. Larger amplitude waves carry more energy. The rate at which energy is transmitted along the string is called power. Power is also related to the amplitude and frequency of the wave.

Applications of Wave Propagation on Strings

The principles of wave propagation on strings have numerous practical applications:

• Musical Instruments: Guitars, violins, pianos, and many other instruments rely on the vibrations of strings to produce sound. The tension, length, and mass of the strings determine the pitch and tone.

• Telecommunications: Fiber optic cables transmit information as light pulses, which can be modeled using similar wave equations to those governing waves on strings.

• Seismic Wave Studies: Understanding wave propagation helps seismologists understand earthquake waves and predict their effects.

• Medical Imaging: Ultrasound technology uses high-frequency sound waves to create images of internal organs.

Advanced Concepts: Damped Waves and Dispersion

While we've focused on ideal scenarios, real-world wave propagation often involves additional factors:

• Damping: Energy loss due to friction and other resistive forces causes waves to lose amplitude over time.

• Dispersion: The speed of a wave can depend on its frequency, leading to the spreading out of a wave pulse.

Conclusion

Understanding wave propagation on a string provides a fundamental base for exploring many areas of physics and engineering. By mastering the concepts of wave speed, frequency, wavelength, standing waves, and their associated equations, you'll be well-equipped to tackle more complex wave phenomena. Remember to practice solving problems, and don't hesitate to explore further resources to deepen your understanding. The examples provided serve as a starting point—experiment with varying parameters to solidify your grasp of these important principles. The world of waves is rich and rewarding, and your journey of discovery is just beginning!