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: 

Recall that for sinusoids of form , the period is given by:

For this problem, the fundamental frequency is when , which means that its frequency is:

The correct answer is  and .

This is because at both waves maxima they would add constructively in the form of  in the positive axis.

When they interfere destructively they would subtract in the form of  in the positive axis.

Therefore the answer is  and .