An important distinction for the MCAT is the difference between transverse and longitudinal waves. Although both wave types are sinusoidal, transverse waves oscillate perpendicular to the direction of propagation, while longitudinal waves oscillate parallel to the direction of propagation.

The most common transverse and longitudinal waves are light waves and sound waves, respectively. All electromagnetic waves (light waves, microwaves, X-rays, radio waves) are transverse. All sound waves are longitudinal.

Transverse waves can be distinguished from longitudinal waves by the orientation of the oscillations to the direction of energy transfer. Transverse waves have oscillations perpendicular to the direction of energy transfer while longitudinal waves have oscillations parallel to the direction of energy transfer. The plucked guitar string may be tricky to think about because we use sound as a characteristic example of a longitudinal wave, and what does a plucked guitar string do but make sound? Well, the sound produced by the string is a longitudinal wave, but the string itself vibrates as a transverse wave. When the string is plucked, the energy is transferred down the string, yet the displacement is up and down or side to side. Meanwhile, an earthquake is a series of compressions that move underground due to shifting along a fault line for one. This is a longitudinal or compression wave. Another example of a transverse wave is those in the ocean. The wave oscillates vertically, causing rises and falls in the water level, but the waves are directed due offshore.

Guitar strings are attached to the guitar at both ends; therefore, each end of the string is a node. From this, we can say that the first harmonic contains only a single antinode. Each time we add another antinode and node (or half of a wave), we reach the next harmonic. We can say that the number of antinodes in the string tells us what harmonic is being played.

The problem statement tells us that the string length is 0.5m and the wavelengths are 0.25meters. This tells us there are two complete waves in the guitar string. This gives us a total of four antinodes; thus we are in the fourth harmonic.

A simple definition of a standing wave is a wave that is self-reinforcing, which is to say that reflection of the wave through the medium results in some areas of amplification (anti-nodes) of the wave and some areas of nullification (nodes). In other words, resonance must occur, and that usually suggests confinement of the wave in some fashion.

A fan and a bus make noise and vibration, but the sound does not resonate. It is transmitted, but not confined. Light with a specific wavelength has no "resonant" character, and neither do waves striking a pier. If the waves were confined in a harbor so that they could amplify, it might be possible to produce a standing wave. Microwaves trapped inside a microwave oven have this feature, producing antinodes of intense heating and nodes where no energy is transmitted into the food; this is the reason that microwave ovens have rotating platforms to make heating of the food item more uniform.

A violin string will be seen to have discrete, stable regions of motion and lack of motion, the requirements of the standing wave phenomenon. The points of reflection on the string are the two ends. The vibration of the wave is confined within the string, amplifying the sound as the nodes overlap.

The equation for frequency of a standing wave on a string is:

This holds true if  takes integer values. When adjusting the value of  to the answer choices, only  will maintain this equality. 

The wavelengths possible for a standing wave on a string with 2 open ends are:

Where  is the length of the string and  is the harmonic given in integers The longest wavelength possible for standing wave occurs when , therefore:

Firstly, since this is an open pipe, the equation for all of its harmonics based on wavelength can be given as:

, where  is the length of the column,  is the wavelength of the wave, and  is the harmonic given in integers. 

Using the relationship between wavelength  and frequency : 

, where  is the wave speed. 

Since we are talking about sound waves in air, we know that its wave velocity is: 

. We also know the column length  is 

We also know that the fundamental frequency occurs when . Therefore:

Frequency of a standing wave  is related to the wave speed and length by: 

, where  is wave speed, and  is the length of the material, and  is the harmonic, given in integers.  

Since the fundamental frequency is given when , 

Plugging in , and 

We start by finding the range of depths allowed by the students' first experiment:

Find the largest time that's within their experimental error:

and the shortest time:

Use these to find the maximum and minimum depths using the kinematics equation:

  is the maximum and

Now we have to use the sound information to get a more precise answer. The well is open at one end and closed at the other, so the resonant wavelengths are given by:

Since the fundamental resonance is a quarter of the wavelength. Find  using the wave equation:

We don't know which of the harmonics, or overtones the students were hearing, so we try the integers until we find a resonant length between  and . For :

Which is way too small. Try the other integers:

Any of these would resonate, but the only one that's within the students' kinematics margin of error corresponds to , so the well must be  deep.

Recall that each additional harmonic will increase the frequency by a factor of , where  is the harmonic number. Conversely, the wavelength will decrease by a factor of . Since we are on the third harmonic , and the length of the string is , the wavelength of the standing wave will be: