Question

A standing wave in a string \(42 \mathrm{cm}\) long has a total of six nodes (including those at the ends). What is the wavelength, in centimeters, of this standing wave?

Short answer

The wavelength of this standing wave is \(16.8 \mathrm{cm}\).

Step-by-step solution

Identifying Given Information

In this problem, we know that the length of the string is \(42 \mathrm{cm}\) and there are a total of six nodes, including the ones at each end.

Understanding Node-Node Distance

In a standing wave, the distance from one node point to the next, often called a node-node distance, is equal to half a wavelength of the wave. So, the wavelength of the wave is twice the node-node distance.

Calculating Node-Node Distance

The node-node distance can be calculated by dividing the total string length by the number of nodes subtracted by one (as the total number of node-node distances will be one less than the total number of nodes). So, the node-node distance (\(D\)) can be calculated using the formula: \(D = \frac{total length}{(total nodes - 1)}\). Substituting the given values into this equation, we get \(D = \frac{42cm}{(6-1)} = 8.4cm \).

Calculating Wavelength

Now that we have the node-node distance, we can calculate the wavelength. Since the wavelength of the wave is twice the node-node distance, we multiply the node-node distance by two to get the wavelength (\(\lambda\)). So, \(\lambda = 2 * D = 2 * 8.4cm = 16.8 cm \).

Key concepts

Nodes in a wave

In a standing wave, nodes are specific points where the amplitude of the wave is always zero. These points are due to destructive interference, where two waves overlap and cancel each other out at these fixed locations. Understanding the role of nodes is essential in analyzing wave patterns.

To visualize a node, imagine a jump rope anchored at two ends, being whipped up and down in rhythm. The nodes are the points on the rope that seem to stand still, while between these fixed points, the rope vibrates the most. This wave behavior happens due to the wave reflecting back and forth, creating patterns of constructive (increased amplitude) and destructive (zero amplitude) interference.

In our exercise, the standing wave on the string has six nodes, including the ones at each end, which influence how the wave behaves and determines the entire pattern along the string.

Wavelength calculation

Wavelength calculation is an essential part of understanding standing waves. Wavelength is defined as the distance between two consecutive points that are in phase along the wave, such as from node to node twice, or crest to crest.

In the given scenario, the total length of the string is 42 centimeters, and there are six nodes. It’s crucial to recognize that the distance between nodes is actually half a wavelength. So, first, we need to determine the node-node distance, which is calculated as the total length of the string divided by the number of node-intervals (or sections between nodes, which is the total number of nodes minus one).

Using the formula:

• Node-node distance (D) = \( \frac{\text{Total length of the string}}{\text{Number of nodes - 1}} \)

Applying this, we find that

• D = \( \frac{42 \text{ cm}}{5} = 8.4 \text{ cm} \)

Next, because we know one wavelength equals twice the node-node distance, we calculate:

• Wavelength (\lambda) = 2 \times 8.4 \text{ cm} = 16.8 \text{ cm}

Wave physics

Wave physics is a broad and fundamental area of physics that explores the behavior, properties, and types of waves, including standing waves. Waves transfer energy through a medium without the net movement of the particles in that medium. In our context, waves are patterns of vibrations that propagate through a medium such as strings, air, or even water.

There are different wave types that occur, such as transverse and longitudinal waves. Standing waves, like the one analyzed in this exercise, occur when two waves of the same frequency and amplitude, traveling in opposite directions, interfere with each other. This phenomenon results in nodes and anti-nodes (points of maximum displacement) along the medium.

Understanding wave physics allows us to not only calculate wavelengths and recognize node configurations, but also to grasp more advanced concepts such as resonance and harmonics. This understanding is crucial in fields ranging from engineering and acoustics to even explaining natural phenomena and technologies we use every day.