Question

At a rock concert, the engineer decides that the music isn't loud enough. He turns up the amplifiers so that the amplitude of the sound, where you're sitting, increases by \(50.0 \%\). (a) By what percentage does the intensity increase? (b) How does the intensity level (in \(\mathrm{dB}\) ) change?

Short answer

(a) The intensity increases by 125%. (b) The intensity level increases by approximately 3.52 dB.

Step-by-step solution

Understand the Relationship Between Amplitude and Intensity

Recall that the intensity of a wave is proportional to the square of its amplitude. That is, if the amplitude increases by a certain percentage, the intensity increases by the square of that percentage increase. Therefore, if the amplitude increases by 50%, it becomes 1.5 times the original amplitude. So, the new intensity \( I_{new} \) is proportional to \((1.5)^2\).

Calculate the Percentage Increase in Intensity

First, calculate the factor by which intensity increases:\[ I_{new} = (1.5)^2 \times I_{original} = 2.25 \times I_{original} \]The intensity has increased by a factor of 2.25. To find the percentage increase in intensity, subtract 1 (the original intensity factor) from 2.25 and multiply by 100%:\[ \\text{Percentage Increase} = (2.25 - 1) \times 100\% = 125\% \]

Calculate the Increase in Intensity Level in Decibels

The intensity level in decibels (dB) is calculated using the formula:\[ \Delta L = 10 \log_{10}\left(\frac{I_{new}}{I_{original}}\right) \]Plugging in the values, we get:\[ \Delta L = 10 \log_{10}(2.25) \]Calculate the logarithm: \[ \Delta L \approx 10 \times 0.352 = 3.52 \text{ dB} \]Thus, the intensity level increases by approximately 3.52 dB.

Key concepts

Amplitude

In the world of sound waves, amplitude is a key player. It's like the height of a wave – the taller the wave, the louder the sound. When you turn up the volume at a rock concert, you're actually increasing the amplitude of the sound waves. This is what we mean when we say the music got louder. The taller the sound wave, the more powerful it is. It's important to understand that amplitude is directly related to the energy of a wave. More amplitude means more energy.

When the engineer increases the amplitude by 50%, the sound waves become 1.5 times their original size. This means that the wave now carries more energy than before. Hence, the listener experiences this increase as the music getting louder. Remember, amplitude isn't just about how big a wave gets; it's about how much energy that wave can deliver.

Decibels

Decibels (dB) are the units used to measure the intensity level of sound. Think of decibels as a way of describing how loud or soft a sound is. In the rock concert scenario, when the sound's amplitude increases, the intensity level also rises. This rise in intensity is measured in decibels, giving us a numerical way to express how much louder the sound becomes.

The formula for calculating the change in sound level in decibels is \( \Delta L = 10 \log_{10}\left(\frac{I_{new}}{I_{original}}\right) \), where \( I_{new} \) is the new intensity and \( I_{original} \) is the original intensity. A change here demonstrates that there's a new energy level being registered by your ears. For instance, at the rock concert, the increase results in a 3.52 dB change. This value tells us precisely how much more intense the sound feels.

Intensity Calculation

Intensity is how much energy a wave carries per unit area. In simpler terms, it's about how powerful the sound is that reaches your ears. When the engineer turns up the volume, it's not just the amplitude that changes but also the intensity. The intensity directly relates to the square of the amplitude.

So, if the amplitude increases by 50%, the intensity doesn't just increase by 50%. It goes up by the square of 1.5 (the factor by which the amplitude increased): \( (1.5)^2 \). This calculation shows that the intensity increases by 225%. If you subtract the original intensity factor (which is 1), you realize that the actual increase is 125% more than the original. This sharp increase makes a big difference in the sound power, explaining that significant jump in loudness.

Wave Properties

Waves are fascinating because they carry energy and information without moving the medium itself from one place to another. The properties of waves include amplitude, wavelength, frequency, and speed. Amplitude, as covered earlier, affects the intensity directly.

Understanding these properties helps explain how sound behaves when it moves through the air. The wavelength and frequency determine the pitch of a sound, while amplitude determines its loudness. Sound travels in waves, and the properties of those waves can change depending on external factors like the environment or a concert sound system.

Knowing these wave properties lets us manipulate sound in practical ways. Whether we're increasing volume at a concert or designing better speakers, understanding wave properties is key. By altering these parameters, engineers can enhance sound performance in different settings.