Business Combinations, Asset Acquisitions, and Joint Venture Formation (Details) € in Millions		3 Months Ended		6 Months Ended
	Sep. 04, 2024 USD ($)	Feb. 28, 2025 USD ($)	Feb. 29, 2024 USD ($)	Feb. 28, 2025 USD ($)	Feb. 29, 2024 USD ($)	Feb. 28, 2025 EUR (€)	Sep. 04, 2024 EUR (€)	Aug. 31, 2024 USD ($)
Business Combination, Asset Acquisition, and Joint Venture Formation [Abstract]
Goodwill, Acquired During Period				$ 14,473,000
Net Sales		$ 145,528,000	$ 138,437,000	290,724,000	$ 280,406,000
Business Acquisition [Line Items]
Net Sales		145,528,000	$ 138,437,000	290,724,000	280,406,000
Goodwill, Acquired During Period				14,473,000
Payments to Acquire Businesses, Net of Cash Acquired				27,196,000	$ 0
Goodwill		$ 277,241,000		$ 277,241,000				$ 269,597,000
Developed Technology
Business Acquisition [Line Items]
Weighted Average Amortization Period (Years)		3 years		3 years		3 years
Customer Relationships
Business Acquisition [Line Items]
Weighted Average Amortization Period (Years)		10 years		10 years		10 years
Technology-Based Intangible Assets
Business Acquisition [Line Items]
Weighted Average Amortization Period (Years)		14 years		14 years		14 years
DTA
Business Combination, Asset Acquisition, and Joint Venture Formation [Abstract]
Business Combination, Contingent Consideration Arrangements, Range of Outcomes, Value, High | €							€ 36.0
Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Property, Plant, and Equipment		$ 2,841,000		$ 2,841,000
Net Sales		3,200,000		6,400,000
Business Acquisition [Line Items]
Net Sales		3,200,000		6,400,000
Business Combination, Contingent Consideration Arrangements, Range of Outcomes, Value, High | €							36.0
Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Property, Plant, and Equipment		2,841,000		2,841,000
Payments to Acquire Businesses, Net of Cash Acquired	$ 27,200,000
Business Combination, Contingent Consideration, Liability | €						€ 2.3	€ 12.0
Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Intangible Assets, Other than Goodwill		14,977,000		14,977,000
Goodwill		14,473,000		14,473,000
DTA | Developed Technology
Business Acquisition [Line Items]
Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Finite-Lived Intangibles		$ 1,600,000		$ 1,600,000
Weighted Average Amortization Period (Years)		3 years		3 years		3 years
DTA | Customer Relationships
Business Acquisition [Line Items]
Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Finite-Lived Intangibles		$ 3,600,000		$ 3,600,000
Weighted Average Amortization Period (Years)		14 years		14 years		14 years
DTA | Technology-Based Intangible Assets
Business Acquisition [Line Items]
Business Combination, Recognized Identifiable Assets Acquired and Liabilities Assumed, Finite-Lived Intangibles		$ 9,800,000		$ 9,800,000
Weighted Average Amortization Period (Years)		14 years		14 years		14 years