Question

Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m, and wavelength 0.320 m. The waves travel in the $-x$-direction, and at $t=0$ the $x=0$ end of the string has its maximum upward displacement. (a) Find the frequency, period, and wave number of these waves. (b) Write a wave function describing the wave. (c) Find the transverse displacement of a particle at $x=0.360$ m at time $t=0.150$ s. (d) How much time must elapse from the instant in part (c) until the particle at $x=0.360$ m next has maximum upward displacement?

Short answer

Frequency: 25 Hz, period: 0.04 s, wave number: 19.63 m⁻¹. Next max at 0.190 s.

Step-by-step solution

Calculate Wave Frequency

The wave speed $v$ is related to the wavelength $\lambda$ and frequency $f$ by the formula:$$v=f\cdot \lambda$$Given the wave speed $v=8.00\text{ m/s}$ and wavelength $\lambda =0.320\text{ m}$, we can solve for the frequency:$$f=\frac{v}{\lambda }=\frac{8.00}{0.320}=25.0\text{ Hz}$$

Calculate Wave Period

The period $T$ of a wave is the inverse of the frequency:$$T=\frac{1}{f}=\frac{1}{25.0}=0.040\text{ s}$$

Calculate Wave Number

The wave number $k$ is given by the formula:$$k=\frac{2\pi }{\lambda }$$Substituting the known wavelength:$$k=\frac{2\pi }{0.320}\approx 19.63{\text{ m}}^{-1}$$

Write the Wave Function

The general form of a wave traveling in the negative $x$-direction is:$$y(x,t)=A\sin (kx+\omega t+\varphi )$$where $A=0.0700\text{ m}$, $\omega =2\pi f=2\pi \times 25.0=50\pi \text{ rad/s}$, and the initial condition specifies maximum upward displacement at $x=0$ and $t=0$, so $\varphi =\frac{\pi }{2}$ (since sine of $\pi /2$ gives maximum value). Thus the wave function becomes:$$y(x,t)=0.0700\sin (19.63x+50\pi t+\frac{\pi }{2})$$

Find Transverse Displacement at Given Position and Time

Substitute $x=0.360\text{ m}$ and $t=0.150\text{ s}$ into the wave function:$$y(0.360,0.150)=0.0700\sin (19.63\times 0.360+50\pi \times 0.150+\frac{\pi }{2})$$Calculating the argument:$$19.63\times 0.360+50\pi \times 0.150+\frac{\pi }{2}=7.0676+23.5619+1.5708\approx 32.2003\text{ rad}$$$$y=0.0700\sin (32.2003)\approx 0.026\text{ m}$$

Calculate Time for Next Maximum Displacement

The time interval $\mathrm{\Delta }t$ between successive maximum displacements is the period $T$, and the sine function reaches its maximum at one complete period later. Thus, from $t=0.150\text{ s}$, the next maximum displacement occurs at:$$t_{\text{next max}}=0.150+0.040=0.190\text{ s}$$Time elapsing from given time to next maximum:$$\mathrm{\Delta }t=0.190-0.150=0.040\text{ s}$$

Key concepts

Transverse Waves

Transverse waves are fascinating waves that move perpendicular to the direction of wave propagation. Imagine holding a rope and moving your hand up and down; the waves travel horizontally while the medium (rope) moves vertically.
This is exactly how transverse waves behave.

• An example of transverse waves can be seen in water waves or electromagnetic waves.
• In the context of waves on a string, the particles of the string move up and down as the wave travels along the string.
• These waves are different from longitudinal waves where the movement of the medium is parallel to wave travel, like sound waves in air.

In our exercise, these transverse waves on the string have a specific wave speed, amplitude, and they travel in the opposite of the positive x-direction, or (-x)-direction, showcasing the versatile nature of how waves can move across mediums.

Wave Speed

Wave speed is a crucial aspect of understanding how fast a wave travels through a medium. It determines how quickly the wave displaces particles along its path.
To find wave speed in a real-world scenario, it's vital to understand the relationship between wave speed, frequency, and wavelength:

• The formula to find wave speed is: $$v=f\cdot \lambda$$Where:
  • $v$ is the wave speed.
  • $f$ is the frequency, or the number of wave cycles per second.
  • $\lambda$ is the wavelength, or the space between repeating units of wave pattern.

In the original problem, we found that the wave speed of the transverse wave on the string is 8.00 m/s. Knowing this makes it easier to calculate related properties like frequency and period using given physical quantities.

Wavelength

Wavelength is a measure of the distance between two identical phases of a wave, such as from crest to crest or trough to trough. It defines the length of a full wave cycle in a particular medium.

• Physically, the wavelength can affect how waves interact with each other and with obstacles.
• In our exercise, the given wavelength is 0.320 m, an important starting parameter. It helps us calculate other properties like the wave number.
• The wavelength also impacts the wave speed and frequency, based on the formula $$v=f\cdot \lambda$$, showing that all these characteristics are intertwined.

By understanding the wavelength, we gain insight into how far a wave must travel before repeating itself, which is essential in wave physics applications.

Wave Function

A wave function is a mathematical description of the wave's oscillation in a medium as it travels. It provides a detailed explanation of displacement over time and space.

• The general form of the wave function is: $$y(x,t)=A\sin (kx+\omega t+\varphi )$$where:
  • $A$ is the amplitude, or maximum displacement.
  • $k$ is the wave number, related to wavelength.
  • $\omega$ is the angular frequency, found from $\omega =2\pi f$.
  • $\varphi$ is the phase constant, accounting for initial displacement.
• For our problem, the wave function quantifies how the wave travels in the negative x-direction: $$y(x,t)=0.0700\sin (19.63x+50\pi t+\frac{\pi }{2})$$
• By using the wave function, we can predict the position of any particle on the string at any time.

Understanding wave functions is fundamental in wave physics since they allow us to model and anticipate wave behaviors naturally observed or in laboratory settings.