كيف يحسب الرقم الهيدروجيني pH للمحاليل؟ How to calculate pH and pOH?

يحسب الرقم الهيدروجيني بالعلاقة: pH = -log[H⁺]، وللمحاليل المائية عند 25°C: pH + pOH = 14. إذا كان pH 7 فالمحلول قاعدي. [الصيغة الرمزية الموزونة: pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴]

Theme C — Wave Behaviour

C.4.3 Boundary Conditions for Standing Waves — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 29 of 37 1 quiz

📚 IB Physics HL Hub / Topical Revision

Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution

pH = -log[H⁺]  |  pOH = -log[OH⁻]  |  pH + pOH = 14  |  [H⁺][OH⁻] = 10⁻¹⁴

Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14

Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.

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Lesson summary

C.4.3 Boundary Conditions for Standing Waves — IB PHYSICS SL notes: Decide whether a boundary imposes a node or an antinode and predict the allowed standing-wave patterns for strings and pipes.Boundary conditions — constraints at the edges of the medium that decide where nodes an…

Allowed patterns
Unsustained waves
Number of nodes
Possible boundaries
String fixed at both ends
Natural frequencies

Part 1 of 7Why boundaries matter

Boundary conditions

Boundary conditions constraints at the edges of the medium that decide where nodes and antinodes must lie

Allowed patterns

Allowed patterns they decide which wavelengths, and so which frequencies, are possible

Media

Media standing waves form on strings and in pipes when a progressive wave superposes with its reflection

Unsustained waves

Unsustained waves waves that fail the conditions are not sustained, because the reflected wave does not reinforce the incident wave

Number of nodes

Number of nodes it depends on the frequency of the progressive waves and on the boundary conditions

Part 2 of 7Strings

1Possible boundaries

Possible boundaries fixed at both ends, free at both ends, or one end fixed and one free

2Fixed end

Fixed end the string cannot move there, so there must be a node
Examples are a string clamped at a wall or the bridge of a guitar.
The reflected wave is in antiphase (a phase change of π) and cancels the incident wave at the end.

3Free end

Free end the end moves freely, so an antinode forms and the wave reflects in phase without inversion
An example is a ring sliding on a rod.

4String fixed at both ends

String fixed at both ends a node lies at each end, so an integer number of half-wavelengths fits the length, L = nλ/2

5Natural frequencies

Natural frequencies at these specific frequencies standing waves with different numbers of nodes and antinodes form

6What sets the frequency

What sets the frequency the tension, adjusted by tuning pegs, and the mass per unit length, which is why guitar strings have different thicknesses

Part 3 of 7Pipes

Possible boundaries

Possible boundaries closed at both ends, open at both ends, or closed at one end and open at the other

Closed end

Closed end it is a displacement node because air cannot move through the stopper or wall
The reflected wave is in antiphase with the incident wave and they cancel.

Open end

Open end it is a displacement antinode because air is free to move
The reflected wave is in phase and the amplitudes add.

End correction

End correction in reality the antinode lies slightly beyond the pipe, but this is ignored unless stated

Pipe open at both ends

Pipe open at both ends antinodes at both ends

Pipe closed at one end

Pipe closed at one end a node at the closed end and an antinode at the open end

Length relation

Length relation a pipe closed at one end has L = λ/4 for its lowest pattern, not λ/2

Part 4 of 7Reflection and phase

1String reflection

String reflection a wave reflected from a fixed end changes phase by π, and there is no change at a free end

2Pressure in a pipe

Pressure in a pipe an open end reflects a compression as a rarefaction, and a closed end reflects a compression as a compression
The open end is a pressure node and the closed end is a pressure antinode.
This is the opposite of the displacement picture.

3Two strings joined

Two strings joined a thicker string acts like a nearly fixed end for the thinner string and inverts the reflected pulse
A thinner string behaves more like a free end.

Part 5 of 7Finding the allowed patterns

Sketch the medium.

Put a node or an antinode at each end according to the boundary type.

Draw the simplest pattern that fits between the ends.

Draw the patterns with additional nodes.

Count how many half-wavelengths or quarter-wavelengths fit and write L as a multiple of them.

Find the wavelength, then the frequency from f = v/λ.

Speed

Speed use about 340 m s⁻¹ for sound in air at room temperature and the given speed for a string

Estimate check

Estimate check a longer string or pipe must give a lower note

Part 6 of 7Resonance in a closed pipe

Driven pipe

Driven pipe a pipe closed at one end driven by a loudspeaker at the open end resonates strongly where the closed end is a node and the open end an antinode

Shortening

Shortening each resonant frequency rises in proportion

Slide whistle

Slide whistle it changes pitch continuously by varying the length of the air column

Part 7 of 7More from the specification

Key terms

Term	Meaning
Boundary condition	Constraint at an edge fixing whether there is a node or antinode.
Fixed end	Point that cannot move, so a node.
Free end	Point free to move, so an antinode.
Open pipe end	Displacement antinode.
Closed pipe end	Displacement node.

Avoid these mistakes

Placing an antinode at a closed end of a pipe — the displacement there is zero.

Forgetting that reflection from a fixed end inverts the wave.

Mixing up pressure nodes and displacement nodes.

Using L = λ/2 for a pipe closed at one end instead of L = λ/4.

Thinking boundary conditions are optional — only waves that fit them persist.

Last updated: 15 October 2026