Question:
A transverse wave given by $y = 2 \sin(0.01x + 30t)$ moves on a stretched string from one end to another end in $0.5\ \text{second}$. If '$x$' and '$y$' are in $\text{cm}$ and '$t$' is in $\text{second}$, then the length of the string is
A transverse wave given by $y = 2 \sin(0.01x + 30t)$ moves on a stretched string from one end to another end in $0.5\ \text{second}$. If '$x$' and '$y$' are in $\text{cm}$ and '$t$' is in $\text{second}$, then the length of the string is
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Always double-check the measurement units specified in wave problems! Here, $x$ is given in centimeters, which means your initial velocity calculation yields $\text{cm/s}$. Skipping the conversion to meters per second can easily lead to a factor-of-100 error.
Updated On: Jun 18, 2026
- $6\ \text{m}$
- $9\ \text{m}$
- $12\ \text{m}$
- $15\ \text{m}$
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
We are given the wave function equation for a progressive transverse wave propagating along a stretched string. The total travel time across the string length is given as $0.5\ \text{seconds}$. We need to compute the physical spatial length of this string.
Step 2: Key Formula or Approach:
1. Compare the given equation with the standard progressive wave formula: $$y = A \sin(kx + \omega t)$$ Where $k$ is the wave number and $\omega$ is the angular frequency. 2. The linear wave propagation speed ($v$) is given by the relation: $$v = \frac{\omega}{k}$$ 3. The total distance covered (length of the string, $L$) in time $t$ is: $$L = v \cdot t$$
Step 3: Detailed Explanation:
From the provided wave equation $y = 2 \sin(0.01x + 30t)$: The wave number coefficient of $x$ is $k = 0.01\ \text{cm}^{-1}$. The angular frequency coefficient of $t$ is $\omega = 30\ \text{rad/s}$. Let's compute the wave velocity $v$: $$v = \frac{\omega}{k} = \frac{30}{0.01} = 3000\ \text{cm/s}$$ Convert this velocity value from centimeters per second to standard SI meters per second units: $$v = \frac{3000}{100} = 30\ \text{m/s}$$ The time taken for the wave front to travel across the entire length of the string is $t = 0.5\ \text{seconds}$. Now compute the length $L$: $$L = v \cdot t = 30 \times 0.5 = 15\ \text{m}$$
Step 4: Final Answer:
The total length of the string is $15\ \text{m}$, which corresponds to option (D).
We are given the wave function equation for a progressive transverse wave propagating along a stretched string. The total travel time across the string length is given as $0.5\ \text{seconds}$. We need to compute the physical spatial length of this string.
Step 2: Key Formula or Approach:
1. Compare the given equation with the standard progressive wave formula: $$y = A \sin(kx + \omega t)$$ Where $k$ is the wave number and $\omega$ is the angular frequency. 2. The linear wave propagation speed ($v$) is given by the relation: $$v = \frac{\omega}{k}$$ 3. The total distance covered (length of the string, $L$) in time $t$ is: $$L = v \cdot t$$
Step 3: Detailed Explanation:
From the provided wave equation $y = 2 \sin(0.01x + 30t)$: The wave number coefficient of $x$ is $k = 0.01\ \text{cm}^{-1}$. The angular frequency coefficient of $t$ is $\omega = 30\ \text{rad/s}$. Let's compute the wave velocity $v$: $$v = \frac{\omega}{k} = \frac{30}{0.01} = 3000\ \text{cm/s}$$ Convert this velocity value from centimeters per second to standard SI meters per second units: $$v = \frac{3000}{100} = 30\ \text{m/s}$$ The time taken for the wave front to travel across the entire length of the string is $t = 0.5\ \text{seconds}$. Now compute the length $L$: $$L = v \cdot t = 30 \times 0.5 = 15\ \text{m}$$
Step 4: Final Answer:
The total length of the string is $15\ \text{m}$, which corresponds to option (D).
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