In order to find the velocity needed, we use the Doppler equation:

Where  is the speed of sound in air,  is the velocity of the observer, and  is the velocity of the source, which is zero in this case since the guitar player is not moving. Since the guitarist is sharp, your adjustment must decrease the perceived frequency by , which means we use the negative sign in the numerator of the Doppler equation. Plug in the given values into the equation and solve for .

Because we need the note to be the final note to be lower in frequency than the original frequency, we need to run away from it.

The speed of sound is fastest in the least compressible media of the lowest density. Sound does not propagate in a vacuum. Air and water are compressible media, so sound does not travel as fast in these as it does in glass, an incompressible medium. In general, the speed of sound is greatest in solids, and within each phase, faster as density decreases.

We simply need to know the Doppler effect equation to solve this problem:

Since the observer (you) is not moving, we can rewrite the equation as:

The reason for using addition in the denominator is explained below.

We know all of our values, so we can simply solve for the perceived frequency:

For the Doppler effect equation, don't waste your time trying to memorize whether the signs should be . We have purposely made them the same to emphasize the following idea. Think about the situation practically. The police car is moving away from you. Therefore, we know that the percieved frequency is going to be less than the source. How do we make the frequency lower? We either lower the numerator or increase the denominator. Since we can only manipulate the denominator, we will use the + sign. Knowing this will also help you immediately eliminate all answers in which the frequency is increased.

This is a two-step problem. First, we need to calculate the percieved frequency that you here, and then convert that frequency into a wavelength.

For calculating percieved frequency, we need the Doppler equation:

Plugging in our values we get:

The reasoning for the signs used is explained below.

Now we need to conver that frequency to a wavelength using the equation:

If you don't know this equation, use your units to reason it out!

Plugging in our values, we get:

For the Doppler effect equation, don't waste your time trying to memorize whether the signs should be . We have purposely made them the same to emphasize the following idea. Think about the situation practically. You are moving away from the source of the sound. This will lower the percieved frequency. How do we lower the frequency? Make the numerator smaller. Therefore, we subtract your (the observer's) velocity. The source is moving toward you. This will increase the percieved frequency. How do we increase the frequency? Decrease the denominator. Therefore, we also subtract his (the source's) velocity.

We need to know the equation for the Doppler effect to solve this problem:

Instead of memorizing the sign conventions, we can think about the situation pratically. You are traveling towards the source, which will increase your perceived frequency; therefore, we'll add your velocity. The source is traveling toward you, which will further increase the perceived frequency. Since the source's velocity is in the denominator, we will subtract it.

Plugging in values for each variable:

We need the Doppler effect equation to solve this problem:

Instead of memorizing sign conventions, we can think about the situation practically. The chasers are traveling toward the storm, increasing their percieved frequency. Since their velocity is in the numerator, we'll add it to the speed of sound. The storm is traveling away from them, decreasing their percieved frequency. That velocity is in the denominator, so we will also add it.

We are solving for actual frequency (not the perceived frequency), so we need to rearrange the equation:

We have values for each variable, allowing us to solve:

Remember the equation for the doppler effect for a moving observer: 

Now, identify the given information:

 (This is the speed of sound in air.)

When you are moving towards the concert, the plus sign is used. Therefore, the apparent frequency is

When you are moving away from the concert, the negative sign is used. Therefore, the apparent frequency is 

The doppler shift of a wave can be caused by the motion of either the observer (the astronomer) or the source (the star). Since the frequency is decreased, the motion must be away, either by the star or the Earth or both.

The difference between the observed frequency and the frequency of the source due to relative velocity is called the Doppler effect, and is given by the following formula:

where  is the observed frequency,  is the source frequency,  is the speed of sound,  is the speed of the observer, and  is the speed of the source. The speed of the observer is equal to zero, because the group of people is stationary. The speed of sound was given to us as . The frequency of the source was given as 421Hz. We must convert the speed of the source into , which when converted is equal to . Finally, we must determine whether the sign in the denominator term should be a plus or minus; because we are approaching the group of people, the observed frequency should be higher, therefore a minus sign is appropriate (makes a smaller denominator which makes a larger result). Our final answer after plugging everything in is 442Hz.

The formula for the doppler effect is:

, where  is the apparent frequency of the wave,  is the actual frequency,  is the wave velocity,  is the velocity of the detector,  is the velocity of the source. In our case, the detector (which is you) is moving towards the source at speed , making the numerator a plus sign. The source is moving away from us at speed , making the denominator also a plus sign. Knowing this, we can now plug into our equation: