Wave speed  is related to wavelength and frequency by:

, where  is wavelength and  is frequency. 

If we wanted to change  by 2, and change both  and  by the same factor , we could set this up as:

If we isolated just the factors by which we're changing the wave speed, 

We'd have to change both wavelength and frequency by 

The equation for the speed of light through materials of different indices of refraction is given by the following equation:

Solve for velocity:

We can calculate the velocity  of a wave if we know the wavelength  and the frequency  of the wave by the following equation,

We are told , and can calculate the frequency 

Therefore, we can find the velocity  of the waves,

The distance between each anti-node in a standing wave is only half a wavelength, so we need to multiply the distance measured between anti-nodes by 2.  Multiply the wavelength by the frequency to get the speed of the wave.

We need to remember to convert our centimeters to meters and our megaHertz to Hertz so the units work out:

We need to utilize two equations for this question. The first of which is:

Here,  is wave speed,  is tension force of the material, and  is linear density. The othe equation we need to use is:

Here,  is wave speed,  is wavelength and  is frequency.

In our case:

plug in known values and solve for wave speed.

Solve the second equation for .

Plug in known values and solve for wavelength.

Sound waves are longitudinal waves, meaning that the waves propagate by compression and rarefaction of their medium. They are termed longitudinal waves because the particles in the medium through which the wave travels (air molecules in our case) oscillate parallel to the direction of motion. Alternatively, transverse waves oscillate perpendicular to the direction of motion. Common examples of transverse waves include light and, to a basic approximation, waves on the ocean.

Longitudinal waves transmit energy by compressing and rarefacting the medium in the same direction as they are traveling. Sounds waves are longitudinal waves and travel by compressing the air through which they travel, causing vibration.

Light, X-rays, and microwaves are all examples of electromagnetic waves; even if you cannot recall if they are longitudinal or transverse, they are all members of the same phenomenon and will have the same type of wave transmission. Transverse waves are generated by oscillation within a plane perpendicular to the direction of motion. Oscillating a rope is a transverse wave, as it is not compressing in the direction of motion.

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.