The amplitude of a sound wave is measured in terms of its maximum gauge pressure. By what factor does the amplitude of a sound wave increase if the sound level goes up by 38.0 dB?
ΔL= L₁ - L₂ =
=10 log₁₀(I₁/I₀) – 10log₁₀(I₂/I₀) =
=10 log₁₀(I₁/I₂)
ΔL/10 = log₁₀(I₁/I₂)
I∼A²
A₁/A₂=sqrt{ I₁/I }=sqrt{10^( ΔL/10)} =
=sqrt{10^( 38/10)} =79.4
To find the factor by which the amplitude of a sound wave increases when the sound level goes up by 38.0 dB, you need to use the formula:
\( \Delta L = 10 \log_{10} \left( \frac{I}{I_0} \right) \)
Where:
- ΔL is the change in sound level (in dB),
- I is the final intensity of the sound wave, and
- I0 is the initial intensity of the sound wave.
In this case, since we want to find the factor by which the amplitude increases, we can rewrite the formula as:
\( \frac{I}{I_0} = 10^{\frac{\Delta L}{10}} \)
Given that the sound level increases by 38.0 dB, we can substitute this value into the formula:
\( \frac{I}{I_0} = 10^{\frac{38.0}{10}} \)
Calculating this expression gives us:
\( \frac{I}{I_0} = 10^{3.8} \)
Using a calculator, we can evaluate this expression to find that:
\( \frac{I}{I_0} \approx 6309.57 \)
Therefore, the amplitude of the sound wave increases by a factor of approximately 6309.57 when the sound level goes up by 38.0 dB.
To determine the factor by which the amplitude of a sound wave increases when the sound level goes up by a certain number of decibels (dB), we need to use the formula:
ΔL = 10 log₁₀ (A/A₀)
where:
ΔL is the change in sound level in dB,
A is the new amplitude,
A₀ is the initial amplitude.
In this case, we know the change in sound level (ΔL) is 38.0 dB. Let's say the factor by which the amplitude increases is "x".
Using the formula, we rearrange it to solve for "x":
38.0 = 10 log₁₀ (xA₀ / A₀)
We can simplify the equation to:
38.0 = 10 log₁₀ (x)
Finally, we isolate "x" by dividing both sides by 10 and converting from logarithmic form to exponential form:
log₁₀ (x) = 3.8
10^(log₁₀ (x)) = 10^(3.8)
x = 10^3.8
Using a calculator, we find that x is approximately 6309.5734.
Therefore, the amplitude of the sound wave increases by a factor of approximately 6309.5734 when the sound level goes up by 38.0 dB.