give answer in 2 step with explanation at the end of each step and final answer at the end:In the diagram below, the blue curve refers ro a sinusoidal wave on a string ar time “= 0 that is propagating

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:In the diagram below, the blue curve refers ro a sinusoidal wave on a string ar time “= 0 that is propagating to
the left. The
red curve represents the same wave ar a later time f= 0.505, before the wave has traveled one full wavelength.
(a) Which of the following wave functions correctly represents the above situation? Here A represents the
amplitude, k
the wave number, and wrhe angular frequency of the wave.
v(x, 1) = Asin(kx - cr)
lx, 1) = Asin(kx + wt)
lx, 1) = Acos(kx - wt)
lx, 1) = Acos(kx + wr)
(b) What is the speed of the wave?
vel

s
(c) What are the amplitude, wave number, and angular frequency of the wave?
A =|, xem
k= What is the wavelength of the wave from the graph? How is the wavelength related to the wave number?
om’
w= x

How is the angular frequency related to the wave number and wave speed? a

VIPSolutions · Accepted Answer

## Step 1: Identify the Correct Wave Function *(Part a)*

### Explanation:
For a wave traveling **to the left**, the general wave function is:
$$ y(x, t) = A\sin(kx + \omega t) $$
or
$$ y(x, t) = A\cos(kx + \omega t) $$
The **plus sign** in $kx + \omega t$ indicates leftward propagation.

**Correct Option:**  
$$
\boxed{y(x, t) = A\sin(kx + \omega t)}
$$

---

## Step 2: Speed, Amplitude, Wave Number, and Angular Frequency *(Parts b & c)*

### (b) Speed of the Wave:
The speed $v$ of a wave is given by:
$$
v = \frac{\omega}{k}
$$
Where:
- $\omega$ = angular frequency
- $k$ = wave number

### (c) Amplitude, Wave Number, and Angular Frequency:
- **Amplitude ($A$)**: Given as 1 cm.
- **Wave Number ($k$)**: 
  $$
  k = \frac{2\pi}{\lambda}
  $$
  Where $\lambda$ is the wavelength (read from graph).
- **Angular Frequency ($\omega$)**: 
  $$
  \omega = 2\pi f
  $$
  Where $f$ is the frequency.

**Relationship:**  
$$
\omega = vk
$$
(Angular frequency equals wave speed times wave number.)

---

## **Final Answers**

### (a) Correct wave function:  
$$
\boxed{y(x, t) = A\sin(kx + \omega t)}
$$
(For a wave moving to the left.)

### (b) Speed of the wave:  
$$
\boxed{v = \frac{\omega}{k}}
$$

### (c)  
- **Amplitude:** $A = 1\, \text{cm}$
- **Wave number:** $k = \frac{2\pi}{\lambda}$
- **Angular frequency:** $\omega = 2\pi f$
- **Relationship:** $\omega = vk$

---

**Summary:**  
- Use $y(x, t) = A\sin(kx + \omega t)$ for leftward movement.
- Speed: $v = \omega/k$.
- Amplitude, wave number, and angular frequency are related through the wave's physical properties as above.

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