Speed on Stretched Strings
When you pluck a guitar string, a wave travels along it at a very specific speed — and that speed depends on two physical properties of the string: how tightly it is stretched and how much mass it has per unit length. These two factors compete against each other. Greater tension pulls the string back towards equilibrium more forcefully, making the wave travel faster. Greater mass per unit length means the string resists acceleration more stubbornly, making the wave travel slower. Understanding this relationship is fundamental to acoustics, musical instrument design, and the broader study of mechanical waves. The speed of a transverse wave on a stretched string is not the same as the speed of the particles in the string vibrating up and down — it is the speed at which the wave pattern itself propagates along the string's length. Think of it like a crowd doing a 'Mexican wave' in a stadium: individual people barely move, but the wave pattern sweeps around the stadium at a completely different speed. The wave speed on a string depends purely on the medium's mechanical properties, not on the frequency or amplitude of the wave — a key principle that applies throughout wave physics. This concept connects directly to why musicians tune instruments by tightening or loosening strings (changing tension) and why thicker, denser strings produce lower-pitched notes even at the same tension. Mastering the wave speed equation for strings also provides an excellent foundation for understanding wave speed in other media, such as sound in air or seismic waves through rock.
- Identifying the two key properties
- Intuition: why does each property matter?
- Deriving the formula by dimensional analysis
- Calculating linear mass density
- Wave speed is independent of frequency
Identifying the two key properties
Two properties of the string determine wave speed. The first is tension T, measured in newtons (N). Tension is the pulling force that the string exerts along its own length when stretched. The second is linear mass density μ (Greek letter 'mu'), measured in kilograms per metre (kg m⁻¹). Linear mass density tells you how much mass is packed into each metre of the string. A thin nylon guitar string has a much smaller μ than a thick steel bass string.
Intuition: why does each property matter?
Imagine the string as a long chain of tiny masses connected by springs. When a portion of the string is displaced sideways, the tension in the string acts like a restoring force, pulling it back and 'informing' the next section that a disturbance has occurred. Higher tension means this restoring force is stronger, so the disturbance propagates faster — just as a stiffer spring transfers a push more rapidly than a loose one. Higher linear density means each small section of string has more inertia, so it takes longer to respond to the incoming force. This is why v increases with T but decreases with μ.
Deriving the formula by dimensional analysis
Even without a full mechanical derivation, dimensional analysis gives the correct formula. Wave speed v has dimensions [m s⁻¹]. Tension T has dimensions [kg m s⁻²] = [N]. Linear density μ has dimensions [kg m⁻¹]. Forming the ratio T/μ gives dimensions [kg m s⁻² / kg m⁻¹] = [m² s⁻²], and taking the square root gives [m s⁻¹]. This confirms that v = √(T/μ) is dimensionally consistent, and a full mechanical derivation (using Newton's second law applied to a small curved element of the string) confirms it is exactly correct with no extra numerical constants.
Calculating linear mass density
Linear mass density μ is not always given directly; often you must calculate it. If a string has total mass m and total length L, then μ = m/L. For example, a 0.80 m guitar string with a mass of 4.0 g has μ = 0.0040 kg / 0.80 m = 0.005 kg m⁻¹. Note that μ is a property of the string's material and thickness — it does not change when you change the tension or the length of the string on the instrument (assuming the string is uniform).
Wave speed is independent of frequency
A crucial point: for a given string under a given tension, all transverse waves travel at the same speed regardless of their frequency or wavelength. This is a property of linear, non-dispersive media. It means that if you double the frequency of a wave on the string (e.g. by pressing a guitar fret), the wavelength halves but the wave speed remains constant (v = fλ). The wave speed changes only when T or μ changes — for example, when a guitarist tightens a tuning peg.
The Key Equation
- v — Wave speed (m s⁻¹): The speed at which the transverse wave pattern travels along the string.
- T — Tension (N (newtons)): The magnitude of the pulling force applied along the string. Must be kept constant for the formula to apply directly.
- \mu — Linear mass density (kg m⁻¹): The mass per unit length of the string. Calculated as μ = m/L where m is the total mass and L is the total length.
Worked Example
Problem. A piano wire of length 1.20 m has a mass of 6.0 g. It is kept under a tension of 800 N. Calculate the speed of a transverse wave on this wire.
Answer. The transverse wave travels along the piano wire at 400 m s⁻¹.
Origins · 1 minMarin Mersenne and how this was discovered
Mersenne was a French theologian, mathematician, and natural philosopher who conducted the first systematic experimental and mathematical study of vibrating strings in the early 17th century. His 'Mersenne's laws', published in his 1636 work Harmonie universelle, described how the frequency of a vibrating string depends on its length, tension, and mass per unit length. These laws directly underpin the modern formula v = √(T/μ), since wave speed, frequency, and wavelength are linked by v = fλ. He measured the speed of sound, studied musical intervals, and corresponded with Descartes, Galileo, and Pascal. Mersenne ran a vast scientific correspondence network from his monastery cell in Paris, effectively acting as a one-man scientific journal before journals existed — he has been called the 'postmaster of Europe's Republic of Letters'.
Related Concepts
- Wave Speed — The general relationship v = fλ connects wave speed to frequency and wavelength, and is used alongside the string formula to find resonant frequencies.
- Standing Waves on Strings — Standing waves form when waves reflect on a fixed string; their frequencies depend directly on wave speed and hence on tension and linear density.
- Speed of Sound in Different Media — Like strings, sound speed in solids and fluids depends on an elastic restoring property divided by inertia — the same physical reasoning extended to 3D media.
- Transverse and Longitudinal Waves — Waves on strings are transverse; understanding the distinction between transverse and longitudinal waves clarifies why the string formula applies only to transverse oscillations.