How do tension and wave speed relate in a string?

Wave speed in a string is directly proportional to the square root of the tension in the string.

In more detail, the speed of a wave travelling through a string, or any other medium, is determined by the properties of that medium. In the case of a string, these properties include its tension and mass per unit length (also known as linear density). The relationship between wave speed (v), tension (T), and linear density (μ) is given by the formula v = √(T/μ). This means that if you increase the tension in the string, the wave speed will also increase, and vice versa.

The reason for this is that tension in a string creates a restoring force when the string is displaced, which is what allows waves to propagate along the string. The greater the tension, the greater this restoring force, and the faster the wave can travel. However, it's important to note that this relationship is not linear, but rather a square root relationship. This means that if you double the tension, the wave speed will increase by a factor of √2, not 2.

On the other hand, the linear density of the string acts to slow down the wave. The more mass there is in a given length of string, the more inertia the wave has to overcome to propagate, and so the slower it travels. This is why the linear density appears in the denominator of the formula: increasing the linear density decreases the wave speed, and vice versa.

IB Physics Tutor Summary: The speed of a wave on a string goes up when you increase the string's tension because higher tension makes the wave move faster. This happens because the formula for wave speed involves the square root of tension divided by the string's mass per unit length. So, if tension goes up, wave speed increases, but it's a square root relationship, not direct doubling.